diff --git a/samtranslator/model/api/api_generator.py b/samtranslator/model/api/api_generator.py
index 260437b41..620ce7bcc 100644
--- a/samtranslator/model/api/api_generator.py
+++ b/samtranslator/model/api/api_generator.py
@@ -223,6 +223,7 @@ def __init__(  # noqa: PLR0913
         policy: Union[dict[str, Any], Intrinsicable[str]] | None = None,
         security_policy: Intrinsicable[str] | None = None,
         endpoint_access_mode: Intrinsicable[str] | None = None,
+        resolved_variables: dict[str, Any] | None = None,
     ):
         """Constructs an API Generator class that generates API Gateway resources
 
@@ -283,6 +284,10 @@ def __init__(  # noqa: PLR0913
         self.policy = policy
         self.security_policy = security_policy
         self.endpoint_access_mode = endpoint_access_mode
+        # Same stage variables as `variables`, but with parameter references resolved. Used only to hash the
+        # deployment logical id; the stage itself keeps `variables` so the emitted template preserves the
+        # customer's intrinsics. Falls back to `variables` when the caller did not resolve them.
+        self.resolved_variables = resolved_variables if resolved_variables is not None else variables
 
     def _construct_rest_api(self) -> ApiGatewayRestApi:  # noqa: PLR0912
         """Constructs and returns the ApiGateway RestApi.
@@ -474,6 +479,7 @@ def _construct_stage(
                 self.domain,
                 redeploy_restapi_parameters,
                 self.always_deploy,
+                stage_variables=self.resolved_variables,
             )
 
         if self.tags is not None:
diff --git a/samtranslator/model/apigateway.py b/samtranslator/model/apigateway.py
index 5d9d44703..11affe37b 100644
--- a/samtranslator/model/apigateway.py
+++ b/samtranslator/model/apigateway.py
@@ -98,7 +98,7 @@ class ApiGatewayDeployment(Resource):
 
     runtime_attrs = {"deployment_id": lambda self: ref(self.logical_id)}
 
-    def make_auto_deployable(
+    def make_auto_deployable(  # noqa: PLR0913
         self,
         stage: ApiGatewayStage,
         openapi_version: Union[dict[str, Any], str] | None = None,
@@ -106,6 +106,8 @@ def make_auto_deployable(
         domain: dict[str, Any] | None = None,
         redeploy_restapi_parameters: Any | None = None,
         always_deploy: bool | None = False,
+        *,
+        stage_variables: dict[str, Any] | None = None,
     ) -> None:
         """
         Sets up the resource such that it will trigger a re-deployment when Swagger changes or always_deploy is true
@@ -116,6 +118,9 @@ def make_auto_deployable(
         :param openapi_version: string containing value of OpenApiVersion flag in the template
         :param domain: Dictionary containing the custom domain configuration for the API
         :param redeploy_restapi_parameters: Dictionary containing the properties for which rest api will be redeployed
+        :param stage_variables: Stage variables with parameter references already resolved, for hashing only. Callers
+            must resolve them, because an unresolved {"Ref": "SomeParameter"} hashes identically for every parameter
+            value. Falls back to the stage's own (possibly unresolved) variables when not supplied.
         """
         if not swagger:
             return
@@ -131,6 +136,16 @@ def make_auto_deployable(
             hash_input.append(str(openapi_version))
         if domain:
             hash_input.append(json.dumps(domain))
+        # Stage variables are applied with an UpdateStage call, which points the stage back at the
+        # deployment it already references. If they are not part of this hash, a variables-only
+        # change reuses the existing deployment: the new variables are never deployed, and any
+        # deployment made outside SAM since the last SAM deployment is reverted.
+        # Only added when set, so templates without stage variables keep their existing hash.
+        # `stage_variables` is the caller-resolved form; the stage's own value is the fallback and may
+        # still hold intrinsics, which hash identically across parameter values.
+        stage_variables = stage_variables if stage_variables is not None else getattr(stage, "Variables", None)
+        if stage_variables:
+            hash_input.append(json.dumps(stage_variables, sort_keys=True))
         function_names = redeploy_restapi_parameters.get("function_names") if redeploy_restapi_parameters else None
         # The deployment logical id is <api logicalId> + "Deployment"
         # The keyword "Deployment" is removed and all the function names associated with api is obtained
diff --git a/samtranslator/model/sam_resources.py b/samtranslator/model/sam_resources.py
index 09145282d..e7c0a1987 100644
--- a/samtranslator/model/sam_resources.py
+++ b/samtranslator/model/sam_resources.py
@@ -1844,6 +1844,14 @@ def to_cloudformation(self, **kwargs) -> list[Resource]:  # type: ignore[no-unty
         self.BinaryMediaTypes = intrinsics_resolver.resolve_parameter_refs(self.BinaryMediaTypes)
         self.Domain = intrinsics_resolver.resolve_parameter_refs(self.Domain)
         self.Auth = intrinsics_resolver.resolve_parameter_refs(self.Auth)
+        # The deployment logical id is hashed from the stage variables, so parameter references have to be
+        # resolved for that hash to change when the deployed value changes: an unresolved {"Ref": "SomeParameter"}
+        # is byte-identical for every parameter value, which would leave the variables-only change undeployed
+        # (see ApiGatewayDeployment.make_auto_deployable). Unlike Domain above, this is deliberately NOT assigned
+        # back to self.Variables: resolve_parameter_refs mutates its argument and inlines the values, and the
+        # emitted AWS::ApiGateway::Stage must keep the customer's intrinsics. Hence a deep copy, used for the
+        # hash only.
+        resolved_variables = intrinsics_resolver.resolve_parameter_refs(copy.deepcopy(self.Variables))
         redeploy_restapi_parameters = kwargs.get("redeploy_restapi_parameters")
         shared_api_usage_plan = kwargs.get("shared_api_usage_plan")
         template_conditions = kwargs.get("conditions")
@@ -1889,6 +1897,7 @@ def to_cloudformation(self, **kwargs) -> list[Resource]:  # type: ignore[no-unty
             policy=self.Policy,
             security_policy=self.SecurityPolicy,
             endpoint_access_mode=self.EndpointAccessMode,
+            resolved_variables=resolved_variables,
         )
 
         generated_resources = api_generator.to_cloudformation(redeploy_restapi_parameters, route53_record_set_groups)
diff --git a/tests/translator/output/api_with_openapi_definition_body_no_flag.json b/tests/translator/output/api_with_openapi_definition_body_no_flag.json
index b7904447c..c6e9e62b7 100644
--- a/tests/translator/output/api_with_openapi_definition_body_no_flag.json
+++ b/tests/translator/output/api_with_openapi_definition_body_no_flag.json
@@ -55,9 +55,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "ExplicitApiDeployment9252467a1e": {
+    "ExplicitApiDeploymente24cb02185": {
       "Properties": {
-        "Description": "RestApi deployment id: 9252467a1edc49ba35cb258640f5e3734cc9fab1",
+        "Description": "RestApi deployment id: e24cb0218583daf7c5e537a4e639dfb3a2971fdf",
         "RestApiId": {
           "Ref": "ExplicitApi"
         },
@@ -70,7 +70,7 @@
         "CacheClusterEnabled": true,
         "CacheClusterSize": "1.6",
         "DeploymentId": {
-          "Ref": "ExplicitApiDeployment9252467a1e"
+          "Ref": "ExplicitApiDeploymente24cb02185"
         },
         "RestApiId": {
           "Ref": "ExplicitApi"
@@ -230,9 +230,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "ServerlessRestApiDeploymentdb4b9da82a": {
+    "ServerlessRestApiDeployment62318f9715": {
       "Properties": {
-        "Description": "RestApi deployment id: db4b9da82adc6031fcd32bf3a4954485464fc009",
+        "Description": "RestApi deployment id: 62318f9715b2b5932b597575f6a4a0eafd0cf7e3",
         "RestApiId": {
           "Ref": "ServerlessRestApi"
         },
@@ -245,7 +245,7 @@
         "CacheClusterEnabled": true,
         "CacheClusterSize": "1.6",
         "DeploymentId": {
-          "Ref": "ServerlessRestApiDeploymentdb4b9da82a"
+          "Ref": "ServerlessRestApiDeployment62318f9715"
         },
         "RestApiId": {
           "Ref": "ServerlessRestApi"
diff --git a/tests/translator/output/api_with_swagger_and_openapi_with_auth.json b/tests/translator/output/api_with_swagger_and_openapi_with_auth.json
index d215e100d..73d5ccb49 100644
--- a/tests/translator/output/api_with_swagger_and_openapi_with_auth.json
+++ b/tests/translator/output/api_with_swagger_and_openapi_with_auth.json
@@ -54,9 +54,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "ExplicitApiDeployment7c4f7dda23": {
+    "ExplicitApiDeployment8e20079165": {
       "Properties": {
-        "Description": "RestApi deployment id: 7c4f7dda23acd71e4a653861510d82ad7809e562",
+        "Description": "RestApi deployment id: 8e20079165a6421b14f28fc3f812a282e9ca3d90",
         "RestApiId": {
           "Ref": "ExplicitApi"
         },
@@ -69,7 +69,7 @@
         "CacheClusterEnabled": true,
         "CacheClusterSize": "1.6",
         "DeploymentId": {
-          "Ref": "ExplicitApiDeployment7c4f7dda23"
+          "Ref": "ExplicitApiDeployment8e20079165"
         },
         "RestApiId": {
           "Ref": "ExplicitApi"
@@ -267,9 +267,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "ServerlessRestApiDeploymente091b9b9a0": {
+    "ServerlessRestApiDeployment26fa62bf10": {
       "Properties": {
-        "Description": "RestApi deployment id: e091b9b9a0d7b8b9db6fee8c4ad295eb98edde08",
+        "Description": "RestApi deployment id: 26fa62bf10190a228c478ca0189e786b42e35acd",
         "RestApiId": {
           "Ref": "ServerlessRestApi"
         },
@@ -282,7 +282,7 @@
         "CacheClusterEnabled": true,
         "CacheClusterSize": "1.6",
         "DeploymentId": {
-          "Ref": "ServerlessRestApiDeploymente091b9b9a0"
+          "Ref": "ServerlessRestApiDeployment26fa62bf10"
         },
         "RestApiId": {
           "Ref": "ServerlessRestApi"
diff --git a/tests/translator/output/aws-cn/api_with_openapi_definition_body_no_flag.json b/tests/translator/output/aws-cn/api_with_openapi_definition_body_no_flag.json
index ca6140b3f..6f4b1f3cb 100644
--- a/tests/translator/output/aws-cn/api_with_openapi_definition_body_no_flag.json
+++ b/tests/translator/output/aws-cn/api_with_openapi_definition_body_no_flag.json
@@ -63,9 +63,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "ExplicitApiDeployment9252467a1e": {
+    "ExplicitApiDeploymente24cb02185": {
       "Properties": {
-        "Description": "RestApi deployment id: 9252467a1edc49ba35cb258640f5e3734cc9fab1",
+        "Description": "RestApi deployment id: e24cb0218583daf7c5e537a4e639dfb3a2971fdf",
         "RestApiId": {
           "Ref": "ExplicitApi"
         },
@@ -78,7 +78,7 @@
         "CacheClusterEnabled": true,
         "CacheClusterSize": "1.6",
         "DeploymentId": {
-          "Ref": "ExplicitApiDeployment9252467a1e"
+          "Ref": "ExplicitApiDeploymente24cb02185"
         },
         "RestApiId": {
           "Ref": "ExplicitApi"
@@ -246,9 +246,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "ServerlessRestApiDeploymente1212668e0": {
+    "ServerlessRestApiDeployment409ed00828": {
       "Properties": {
-        "Description": "RestApi deployment id: e1212668e096994ab32167666f5a877bd6ac5fad",
+        "Description": "RestApi deployment id: 409ed008280bb6489cdd1707ff7f2b66d5ad40bb",
         "RestApiId": {
           "Ref": "ServerlessRestApi"
         },
@@ -261,7 +261,7 @@
         "CacheClusterEnabled": true,
         "CacheClusterSize": "1.6",
         "DeploymentId": {
-          "Ref": "ServerlessRestApiDeploymente1212668e0"
+          "Ref": "ServerlessRestApiDeployment409ed00828"
         },
         "RestApiId": {
           "Ref": "ServerlessRestApi"
diff --git a/tests/translator/output/aws-cn/api_with_swagger_and_openapi_with_auth.json b/tests/translator/output/aws-cn/api_with_swagger_and_openapi_with_auth.json
index 246a43aef..92be58e11 100644
--- a/tests/translator/output/aws-cn/api_with_swagger_and_openapi_with_auth.json
+++ b/tests/translator/output/aws-cn/api_with_swagger_and_openapi_with_auth.json
@@ -62,9 +62,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "ExplicitApiDeployment7c4f7dda23": {
+    "ExplicitApiDeployment8e20079165": {
       "Properties": {
-        "Description": "RestApi deployment id: 7c4f7dda23acd71e4a653861510d82ad7809e562",
+        "Description": "RestApi deployment id: 8e20079165a6421b14f28fc3f812a282e9ca3d90",
         "RestApiId": {
           "Ref": "ExplicitApi"
         },
@@ -77,7 +77,7 @@
         "CacheClusterEnabled": true,
         "CacheClusterSize": "1.6",
         "DeploymentId": {
-          "Ref": "ExplicitApiDeployment7c4f7dda23"
+          "Ref": "ExplicitApiDeployment8e20079165"
         },
         "RestApiId": {
           "Ref": "ExplicitApi"
@@ -283,9 +283,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "ServerlessRestApiDeployment614ec93e15": {
+    "ServerlessRestApiDeploymente8a33c953e": {
       "Properties": {
-        "Description": "RestApi deployment id: 614ec93e159f50797f73789c416660484070ee2e",
+        "Description": "RestApi deployment id: e8a33c953ef3abf412bc61a49eecc59a9eb71bc8",
         "RestApiId": {
           "Ref": "ServerlessRestApi"
         },
@@ -298,7 +298,7 @@
         "CacheClusterEnabled": true,
         "CacheClusterSize": "1.6",
         "DeploymentId": {
-          "Ref": "ServerlessRestApiDeployment614ec93e15"
+          "Ref": "ServerlessRestApiDeploymente8a33c953e"
         },
         "RestApiId": {
           "Ref": "ServerlessRestApi"
diff --git a/tests/translator/output/aws-cn/explicit_api.json b/tests/translator/output/aws-cn/explicit_api.json
index 63eefda8a..e9c738301 100644
--- a/tests/translator/output/aws-cn/explicit_api.json
+++ b/tests/translator/output/aws-cn/explicit_api.json
@@ -69,9 +69,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "GetHtmlApiDeploymentf117c932f7": {
+    "GetHtmlApiDeploymentb28d6f4807": {
       "Properties": {
-        "Description": "RestApi deployment id: f117c932f75cfa87d23dfed64e9430d0081ef289",
+        "Description": "RestApi deployment id: b28d6f4807cb961a23c29eeed9b708cab7c109f0",
         "RestApiId": {
           "Ref": "GetHtmlApi"
         },
@@ -82,7 +82,7 @@
     "GetHtmlApiStage": {
       "Properties": {
         "DeploymentId": {
-          "Ref": "GetHtmlApiDeploymentf117c932f7"
+          "Ref": "GetHtmlApiDeploymentb28d6f4807"
         },
         "RestApiId": {
           "Ref": "GetHtmlApi"
diff --git a/tests/translator/output/aws-cn/explicit_api_openapi_3.json b/tests/translator/output/aws-cn/explicit_api_openapi_3.json
index 36f676857..65671de7f 100644
--- a/tests/translator/output/aws-cn/explicit_api_openapi_3.json
+++ b/tests/translator/output/aws-cn/explicit_api_openapi_3.json
@@ -68,9 +68,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "GetHtmlApiDeploymentf117c932f7": {
+    "GetHtmlApiDeploymentb28d6f4807": {
       "Properties": {
-        "Description": "RestApi deployment id: f117c932f75cfa87d23dfed64e9430d0081ef289",
+        "Description": "RestApi deployment id: b28d6f4807cb961a23c29eeed9b708cab7c109f0",
         "RestApiId": {
           "Ref": "GetHtmlApi"
         },
@@ -81,7 +81,7 @@
     "GetHtmlApiStage": {
       "Properties": {
         "DeploymentId": {
-          "Ref": "GetHtmlApiDeploymentf117c932f7"
+          "Ref": "GetHtmlApiDeploymentb28d6f4807"
         },
         "RestApiId": {
           "Ref": "GetHtmlApi"
diff --git a/tests/translator/output/aws-cn/function_with_alias_and_event_sources.json b/tests/translator/output/aws-cn/function_with_alias_and_event_sources.json
index b61186ff1..45c4830d4 100644
--- a/tests/translator/output/aws-cn/function_with_alias_and_event_sources.json
+++ b/tests/translator/output/aws-cn/function_with_alias_and_event_sources.json
@@ -24,9 +24,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "GetHtmlApiDeploymentf117c932f7": {
+    "GetHtmlApiDeployment2acb558823": {
       "Properties": {
-        "Description": "RestApi deployment id: f117c932f75cfa87d23dfed64e9430d0081ef289",
+        "Description": "RestApi deployment id: 2acb558823d4068b07590e8cd40633b8597a3157",
         "RestApiId": {
           "Ref": "GetHtmlApi"
         },
@@ -37,7 +37,7 @@
     "GetHtmlApiStage": {
       "Properties": {
         "DeploymentId": {
-          "Ref": "GetHtmlApiDeploymentf117c932f7"
+          "Ref": "GetHtmlApiDeployment2acb558823"
         },
         "RestApiId": {
           "Ref": "GetHtmlApi"
diff --git a/tests/translator/output/aws-cn/global_handle_path_level_parameter.json b/tests/translator/output/aws-cn/global_handle_path_level_parameter.json
index c52903753..180c9aa90 100644
--- a/tests/translator/output/aws-cn/global_handle_path_level_parameter.json
+++ b/tests/translator/output/aws-cn/global_handle_path_level_parameter.json
@@ -70,9 +70,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "ExplicitApiDeployment9a254aa466": {
+    "ExplicitApiDeployment31e389dd75": {
       "Properties": {
-        "Description": "RestApi deployment id: 9a254aa466c6f818951dfb6e45fde65489beb153",
+        "Description": "RestApi deployment id: 31e389dd75bba46beb7a57c4256c48c5a9b127f4",
         "RestApiId": {
           "Ref": "ExplicitApi"
         },
@@ -85,7 +85,7 @@
         "CacheClusterEnabled": true,
         "CacheClusterSize": "1.6",
         "DeploymentId": {
-          "Ref": "ExplicitApiDeployment9a254aa466"
+          "Ref": "ExplicitApiDeployment31e389dd75"
         },
         "RestApiId": {
           "Ref": "ExplicitApi"
@@ -253,9 +253,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "ServerlessRestApiDeploymente1212668e0": {
+    "ServerlessRestApiDeployment409ed00828": {
       "Properties": {
-        "Description": "RestApi deployment id: e1212668e096994ab32167666f5a877bd6ac5fad",
+        "Description": "RestApi deployment id: 409ed008280bb6489cdd1707ff7f2b66d5ad40bb",
         "RestApiId": {
           "Ref": "ServerlessRestApi"
         },
@@ -268,7 +268,7 @@
         "CacheClusterEnabled": true,
         "CacheClusterSize": "1.6",
         "DeploymentId": {
-          "Ref": "ServerlessRestApiDeploymente1212668e0"
+          "Ref": "ServerlessRestApiDeployment409ed00828"
         },
         "RestApiId": {
           "Ref": "ServerlessRestApi"
diff --git a/tests/translator/output/aws-cn/globals_for_api.json b/tests/translator/output/aws-cn/globals_for_api.json
index ee86381d3..eff4c0773 100644
--- a/tests/translator/output/aws-cn/globals_for_api.json
+++ b/tests/translator/output/aws-cn/globals_for_api.json
@@ -69,9 +69,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "ExplicitApiDeployment43e01e673d": {
+    "ExplicitApiDeployment0b29d48047": {
       "Properties": {
-        "Description": "RestApi deployment id: 43e01e673d7acbd09e4c38ff78dd6ddaf2ed1d55",
+        "Description": "RestApi deployment id: 0b29d480473c478bb1a2337fdc7ff0b80bbe48cd",
         "RestApiId": {
           "Ref": "ExplicitApi"
         },
@@ -84,7 +84,7 @@
         "CacheClusterEnabled": true,
         "CacheClusterSize": "1.6",
         "DeploymentId": {
-          "Ref": "ExplicitApiDeployment43e01e673d"
+          "Ref": "ExplicitApiDeployment0b29d48047"
         },
         "RestApiId": {
           "Ref": "ExplicitApi"
@@ -260,9 +260,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "ServerlessRestApiDeploymentf6c326a165": {
+    "ServerlessRestApiDeployment3556755163": {
       "Properties": {
-        "Description": "RestApi deployment id: f6c326a1656cc9fbb0106cc645598d88575554eb",
+        "Description": "RestApi deployment id: 3556755163aac22f1710a4c6babf8de8932e55b6",
         "RestApiId": {
           "Ref": "ServerlessRestApi"
         },
@@ -275,7 +275,7 @@
         "CacheClusterEnabled": true,
         "CacheClusterSize": "1.6",
         "DeploymentId": {
-          "Ref": "ServerlessRestApiDeploymentf6c326a165"
+          "Ref": "ServerlessRestApiDeployment3556755163"
         },
         "RestApiId": {
           "Ref": "ServerlessRestApi"
diff --git a/tests/translator/output/aws-cn/intrinsic_functions.json b/tests/translator/output/aws-cn/intrinsic_functions.json
index 7a93d4676..b91054164 100644
--- a/tests/translator/output/aws-cn/intrinsic_functions.json
+++ b/tests/translator/output/aws-cn/intrinsic_functions.json
@@ -259,9 +259,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "MyExplicitApiDeployment7145dd00ce": {
+    "MyExplicitApiDeployment011ab6fdbf": {
       "Properties": {
-        "Description": "RestApi deployment id: 7145dd00cea59b4a62b4d7855add490c587f3f62",
+        "Description": "RestApi deployment id: 011ab6fdbfec2a0b749690c4c389b4f44e51a053",
         "RestApiId": {
           "Ref": "MyExplicitApi"
         },
@@ -272,7 +272,7 @@
     "MyExplicitApidevStage": {
       "Properties": {
         "DeploymentId": {
-          "Ref": "MyExplicitApiDeployment7145dd00ce"
+          "Ref": "MyExplicitApiDeployment011ab6fdbf"
         },
         "RestApiId": {
           "Ref": "MyExplicitApi"
diff --git a/tests/translator/output/aws-us-gov/api_with_openapi_definition_body_no_flag.json b/tests/translator/output/aws-us-gov/api_with_openapi_definition_body_no_flag.json
index d95ca5396..c3086b175 100644
--- a/tests/translator/output/aws-us-gov/api_with_openapi_definition_body_no_flag.json
+++ b/tests/translator/output/aws-us-gov/api_with_openapi_definition_body_no_flag.json
@@ -63,9 +63,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "ExplicitApiDeployment9252467a1e": {
+    "ExplicitApiDeploymente24cb02185": {
       "Properties": {
-        "Description": "RestApi deployment id: 9252467a1edc49ba35cb258640f5e3734cc9fab1",
+        "Description": "RestApi deployment id: e24cb0218583daf7c5e537a4e639dfb3a2971fdf",
         "RestApiId": {
           "Ref": "ExplicitApi"
         },
@@ -78,7 +78,7 @@
         "CacheClusterEnabled": true,
         "CacheClusterSize": "1.6",
         "DeploymentId": {
-          "Ref": "ExplicitApiDeployment9252467a1e"
+          "Ref": "ExplicitApiDeploymente24cb02185"
         },
         "RestApiId": {
           "Ref": "ExplicitApi"
@@ -246,9 +246,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "ServerlessRestApiDeploymentc969c99f9d": {
+    "ServerlessRestApiDeployment6ba28851eb": {
       "Properties": {
-        "Description": "RestApi deployment id: c969c99f9d6b6921dff605a206e8989bdb7d1bc7",
+        "Description": "RestApi deployment id: 6ba28851eb50150f1e4349e33bc5185400c063d9",
         "RestApiId": {
           "Ref": "ServerlessRestApi"
         },
@@ -261,7 +261,7 @@
         "CacheClusterEnabled": true,
         "CacheClusterSize": "1.6",
         "DeploymentId": {
-          "Ref": "ServerlessRestApiDeploymentc969c99f9d"
+          "Ref": "ServerlessRestApiDeployment6ba28851eb"
         },
         "RestApiId": {
           "Ref": "ServerlessRestApi"
diff --git a/tests/translator/output/aws-us-gov/api_with_swagger_and_openapi_with_auth.json b/tests/translator/output/aws-us-gov/api_with_swagger_and_openapi_with_auth.json
index 103de4247..3648f6d17 100644
--- a/tests/translator/output/aws-us-gov/api_with_swagger_and_openapi_with_auth.json
+++ b/tests/translator/output/aws-us-gov/api_with_swagger_and_openapi_with_auth.json
@@ -62,9 +62,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "ExplicitApiDeployment7c4f7dda23": {
+    "ExplicitApiDeployment8e20079165": {
       "Properties": {
-        "Description": "RestApi deployment id: 7c4f7dda23acd71e4a653861510d82ad7809e562",
+        "Description": "RestApi deployment id: 8e20079165a6421b14f28fc3f812a282e9ca3d90",
         "RestApiId": {
           "Ref": "ExplicitApi"
         },
@@ -77,7 +77,7 @@
         "CacheClusterEnabled": true,
         "CacheClusterSize": "1.6",
         "DeploymentId": {
-          "Ref": "ExplicitApiDeployment7c4f7dda23"
+          "Ref": "ExplicitApiDeployment8e20079165"
         },
         "RestApiId": {
           "Ref": "ExplicitApi"
@@ -283,9 +283,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "ServerlessRestApiDeployment05a3d3687d": {
+    "ServerlessRestApiDeployment9d5c248b84": {
       "Properties": {
-        "Description": "RestApi deployment id: 05a3d3687d4bfb804b237d40817879c6559ac61c",
+        "Description": "RestApi deployment id: 9d5c248b84fd7e0c6d26463880550ef124b0e145",
         "RestApiId": {
           "Ref": "ServerlessRestApi"
         },
@@ -298,7 +298,7 @@
         "CacheClusterEnabled": true,
         "CacheClusterSize": "1.6",
         "DeploymentId": {
-          "Ref": "ServerlessRestApiDeployment05a3d3687d"
+          "Ref": "ServerlessRestApiDeployment9d5c248b84"
         },
         "RestApiId": {
           "Ref": "ServerlessRestApi"
diff --git a/tests/translator/output/aws-us-gov/explicit_api.json b/tests/translator/output/aws-us-gov/explicit_api.json
index 14ff0657f..0b804b909 100644
--- a/tests/translator/output/aws-us-gov/explicit_api.json
+++ b/tests/translator/output/aws-us-gov/explicit_api.json
@@ -69,9 +69,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "GetHtmlApiDeploymentf117c932f7": {
+    "GetHtmlApiDeploymentb28d6f4807": {
       "Properties": {
-        "Description": "RestApi deployment id: f117c932f75cfa87d23dfed64e9430d0081ef289",
+        "Description": "RestApi deployment id: b28d6f4807cb961a23c29eeed9b708cab7c109f0",
         "RestApiId": {
           "Ref": "GetHtmlApi"
         },
@@ -82,7 +82,7 @@
     "GetHtmlApiStage": {
       "Properties": {
         "DeploymentId": {
-          "Ref": "GetHtmlApiDeploymentf117c932f7"
+          "Ref": "GetHtmlApiDeploymentb28d6f4807"
         },
         "RestApiId": {
           "Ref": "GetHtmlApi"
diff --git a/tests/translator/output/aws-us-gov/explicit_api_openapi_3.json b/tests/translator/output/aws-us-gov/explicit_api_openapi_3.json
index 5383290e6..db090550a 100644
--- a/tests/translator/output/aws-us-gov/explicit_api_openapi_3.json
+++ b/tests/translator/output/aws-us-gov/explicit_api_openapi_3.json
@@ -68,9 +68,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "GetHtmlApiDeploymentf117c932f7": {
+    "GetHtmlApiDeploymentb28d6f4807": {
       "Properties": {
-        "Description": "RestApi deployment id: f117c932f75cfa87d23dfed64e9430d0081ef289",
+        "Description": "RestApi deployment id: b28d6f4807cb961a23c29eeed9b708cab7c109f0",
         "RestApiId": {
           "Ref": "GetHtmlApi"
         },
@@ -81,7 +81,7 @@
     "GetHtmlApiStage": {
       "Properties": {
         "DeploymentId": {
-          "Ref": "GetHtmlApiDeploymentf117c932f7"
+          "Ref": "GetHtmlApiDeploymentb28d6f4807"
         },
         "RestApiId": {
           "Ref": "GetHtmlApi"
diff --git a/tests/translator/output/aws-us-gov/function_with_alias_and_event_sources.json b/tests/translator/output/aws-us-gov/function_with_alias_and_event_sources.json
index 35a2609b2..585ac2cd1 100644
--- a/tests/translator/output/aws-us-gov/function_with_alias_and_event_sources.json
+++ b/tests/translator/output/aws-us-gov/function_with_alias_and_event_sources.json
@@ -24,9 +24,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "GetHtmlApiDeploymentf117c932f7": {
+    "GetHtmlApiDeployment2acb558823": {
       "Properties": {
-        "Description": "RestApi deployment id: f117c932f75cfa87d23dfed64e9430d0081ef289",
+        "Description": "RestApi deployment id: 2acb558823d4068b07590e8cd40633b8597a3157",
         "RestApiId": {
           "Ref": "GetHtmlApi"
         },
@@ -37,7 +37,7 @@
     "GetHtmlApiStage": {
       "Properties": {
         "DeploymentId": {
-          "Ref": "GetHtmlApiDeploymentf117c932f7"
+          "Ref": "GetHtmlApiDeployment2acb558823"
         },
         "RestApiId": {
           "Ref": "GetHtmlApi"
diff --git a/tests/translator/output/aws-us-gov/global_handle_path_level_parameter.json b/tests/translator/output/aws-us-gov/global_handle_path_level_parameter.json
index 633d47f15..82127cdf2 100644
--- a/tests/translator/output/aws-us-gov/global_handle_path_level_parameter.json
+++ b/tests/translator/output/aws-us-gov/global_handle_path_level_parameter.json
@@ -70,9 +70,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "ExplicitApiDeployment9a254aa466": {
+    "ExplicitApiDeployment31e389dd75": {
       "Properties": {
-        "Description": "RestApi deployment id: 9a254aa466c6f818951dfb6e45fde65489beb153",
+        "Description": "RestApi deployment id: 31e389dd75bba46beb7a57c4256c48c5a9b127f4",
         "RestApiId": {
           "Ref": "ExplicitApi"
         },
@@ -85,7 +85,7 @@
         "CacheClusterEnabled": true,
         "CacheClusterSize": "1.6",
         "DeploymentId": {
-          "Ref": "ExplicitApiDeployment9a254aa466"
+          "Ref": "ExplicitApiDeployment31e389dd75"
         },
         "RestApiId": {
           "Ref": "ExplicitApi"
@@ -253,9 +253,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "ServerlessRestApiDeploymentc969c99f9d": {
+    "ServerlessRestApiDeployment6ba28851eb": {
       "Properties": {
-        "Description": "RestApi deployment id: c969c99f9d6b6921dff605a206e8989bdb7d1bc7",
+        "Description": "RestApi deployment id: 6ba28851eb50150f1e4349e33bc5185400c063d9",
         "RestApiId": {
           "Ref": "ServerlessRestApi"
         },
@@ -268,7 +268,7 @@
         "CacheClusterEnabled": true,
         "CacheClusterSize": "1.6",
         "DeploymentId": {
-          "Ref": "ServerlessRestApiDeploymentc969c99f9d"
+          "Ref": "ServerlessRestApiDeployment6ba28851eb"
         },
         "RestApiId": {
           "Ref": "ServerlessRestApi"
diff --git a/tests/translator/output/aws-us-gov/globals_for_api.json b/tests/translator/output/aws-us-gov/globals_for_api.json
index 0f1844e2c..4edd571c5 100644
--- a/tests/translator/output/aws-us-gov/globals_for_api.json
+++ b/tests/translator/output/aws-us-gov/globals_for_api.json
@@ -69,9 +69,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "ExplicitApiDeployment43e01e673d": {
+    "ExplicitApiDeployment0b29d48047": {
       "Properties": {
-        "Description": "RestApi deployment id: 43e01e673d7acbd09e4c38ff78dd6ddaf2ed1d55",
+        "Description": "RestApi deployment id: 0b29d480473c478bb1a2337fdc7ff0b80bbe48cd",
         "RestApiId": {
           "Ref": "ExplicitApi"
         },
@@ -84,7 +84,7 @@
         "CacheClusterEnabled": true,
         "CacheClusterSize": "1.6",
         "DeploymentId": {
-          "Ref": "ExplicitApiDeployment43e01e673d"
+          "Ref": "ExplicitApiDeployment0b29d48047"
         },
         "RestApiId": {
           "Ref": "ExplicitApi"
@@ -260,9 +260,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "ServerlessRestApiDeployment6fd1928d9b": {
+    "ServerlessRestApiDeployment0a9707dedb": {
       "Properties": {
-        "Description": "RestApi deployment id: 6fd1928d9b9ad3c711a371e1337306458029f8bd",
+        "Description": "RestApi deployment id: 0a9707dedb5b0b282252fb37d74c2803a4ea3d1e",
         "RestApiId": {
           "Ref": "ServerlessRestApi"
         },
@@ -275,7 +275,7 @@
         "CacheClusterEnabled": true,
         "CacheClusterSize": "1.6",
         "DeploymentId": {
-          "Ref": "ServerlessRestApiDeployment6fd1928d9b"
+          "Ref": "ServerlessRestApiDeployment0a9707dedb"
         },
         "RestApiId": {
           "Ref": "ServerlessRestApi"
diff --git a/tests/translator/output/aws-us-gov/intrinsic_functions.json b/tests/translator/output/aws-us-gov/intrinsic_functions.json
index b2b44466b..e5159d8af 100644
--- a/tests/translator/output/aws-us-gov/intrinsic_functions.json
+++ b/tests/translator/output/aws-us-gov/intrinsic_functions.json
@@ -259,9 +259,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "MyExplicitApiDeployment7145dd00ce": {
+    "MyExplicitApiDeployment011ab6fdbf": {
       "Properties": {
-        "Description": "RestApi deployment id: 7145dd00cea59b4a62b4d7855add490c587f3f62",
+        "Description": "RestApi deployment id: 011ab6fdbfec2a0b749690c4c389b4f44e51a053",
         "RestApiId": {
           "Ref": "MyExplicitApi"
         },
@@ -272,7 +272,7 @@
     "MyExplicitApidevStage": {
       "Properties": {
         "DeploymentId": {
-          "Ref": "MyExplicitApiDeployment7145dd00ce"
+          "Ref": "MyExplicitApiDeployment011ab6fdbf"
         },
         "RestApiId": {
           "Ref": "MyExplicitApi"
diff --git a/tests/translator/output/explicit_api.json b/tests/translator/output/explicit_api.json
index 95c019102..fb03c8695 100644
--- a/tests/translator/output/explicit_api.json
+++ b/tests/translator/output/explicit_api.json
@@ -53,9 +53,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "GetHtmlApiDeploymentf117c932f7": {
+    "GetHtmlApiDeploymentb28d6f4807": {
       "Properties": {
-        "Description": "RestApi deployment id: f117c932f75cfa87d23dfed64e9430d0081ef289",
+        "Description": "RestApi deployment id: b28d6f4807cb961a23c29eeed9b708cab7c109f0",
         "RestApiId": {
           "Ref": "GetHtmlApi"
         },
@@ -66,7 +66,7 @@
     "GetHtmlApiStage": {
       "Properties": {
         "DeploymentId": {
-          "Ref": "GetHtmlApiDeploymentf117c932f7"
+          "Ref": "GetHtmlApiDeploymentb28d6f4807"
         },
         "RestApiId": {
           "Ref": "GetHtmlApi"
diff --git a/tests/translator/output/explicit_api_openapi_3.json b/tests/translator/output/explicit_api_openapi_3.json
index 5aac47a16..09263bcdb 100644
--- a/tests/translator/output/explicit_api_openapi_3.json
+++ b/tests/translator/output/explicit_api_openapi_3.json
@@ -52,9 +52,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "GetHtmlApiDeploymentf117c932f7": {
+    "GetHtmlApiDeploymentb28d6f4807": {
       "Properties": {
-        "Description": "RestApi deployment id: f117c932f75cfa87d23dfed64e9430d0081ef289",
+        "Description": "RestApi deployment id: b28d6f4807cb961a23c29eeed9b708cab7c109f0",
         "RestApiId": {
           "Ref": "GetHtmlApi"
         },
@@ -65,7 +65,7 @@
     "GetHtmlApiStage": {
       "Properties": {
         "DeploymentId": {
-          "Ref": "GetHtmlApiDeploymentf117c932f7"
+          "Ref": "GetHtmlApiDeploymentb28d6f4807"
         },
         "RestApiId": {
           "Ref": "GetHtmlApi"
diff --git a/tests/translator/output/function_with_alias_and_event_sources.json b/tests/translator/output/function_with_alias_and_event_sources.json
index 574336b38..792b00b41 100644
--- a/tests/translator/output/function_with_alias_and_event_sources.json
+++ b/tests/translator/output/function_with_alias_and_event_sources.json
@@ -16,9 +16,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "GetHtmlApiDeploymentf117c932f7": {
+    "GetHtmlApiDeployment2acb558823": {
       "Properties": {
-        "Description": "RestApi deployment id: f117c932f75cfa87d23dfed64e9430d0081ef289",
+        "Description": "RestApi deployment id: 2acb558823d4068b07590e8cd40633b8597a3157",
         "RestApiId": {
           "Ref": "GetHtmlApi"
         },
@@ -29,7 +29,7 @@
     "GetHtmlApiStage": {
       "Properties": {
         "DeploymentId": {
-          "Ref": "GetHtmlApiDeploymentf117c932f7"
+          "Ref": "GetHtmlApiDeployment2acb558823"
         },
         "RestApiId": {
           "Ref": "GetHtmlApi"
diff --git a/tests/translator/output/global_handle_path_level_parameter.json b/tests/translator/output/global_handle_path_level_parameter.json
index 1eed92346..406eeebe8 100644
--- a/tests/translator/output/global_handle_path_level_parameter.json
+++ b/tests/translator/output/global_handle_path_level_parameter.json
@@ -62,9 +62,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "ExplicitApiDeployment9a254aa466": {
+    "ExplicitApiDeployment31e389dd75": {
       "Properties": {
-        "Description": "RestApi deployment id: 9a254aa466c6f818951dfb6e45fde65489beb153",
+        "Description": "RestApi deployment id: 31e389dd75bba46beb7a57c4256c48c5a9b127f4",
         "RestApiId": {
           "Ref": "ExplicitApi"
         },
@@ -77,7 +77,7 @@
         "CacheClusterEnabled": true,
         "CacheClusterSize": "1.6",
         "DeploymentId": {
-          "Ref": "ExplicitApiDeployment9a254aa466"
+          "Ref": "ExplicitApiDeployment31e389dd75"
         },
         "RestApiId": {
           "Ref": "ExplicitApi"
@@ -237,9 +237,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "ServerlessRestApiDeploymentdb4b9da82a": {
+    "ServerlessRestApiDeployment62318f9715": {
       "Properties": {
-        "Description": "RestApi deployment id: db4b9da82adc6031fcd32bf3a4954485464fc009",
+        "Description": "RestApi deployment id: 62318f9715b2b5932b597575f6a4a0eafd0cf7e3",
         "RestApiId": {
           "Ref": "ServerlessRestApi"
         },
@@ -252,7 +252,7 @@
         "CacheClusterEnabled": true,
         "CacheClusterSize": "1.6",
         "DeploymentId": {
-          "Ref": "ServerlessRestApiDeploymentdb4b9da82a"
+          "Ref": "ServerlessRestApiDeployment62318f9715"
         },
         "RestApiId": {
           "Ref": "ServerlessRestApi"
diff --git a/tests/translator/output/globals_for_api.json b/tests/translator/output/globals_for_api.json
index 6c624d7f4..5ed252057 100644
--- a/tests/translator/output/globals_for_api.json
+++ b/tests/translator/output/globals_for_api.json
@@ -61,9 +61,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "ExplicitApiDeployment43e01e673d": {
+    "ExplicitApiDeployment0b29d48047": {
       "Properties": {
-        "Description": "RestApi deployment id: 43e01e673d7acbd09e4c38ff78dd6ddaf2ed1d55",
+        "Description": "RestApi deployment id: 0b29d480473c478bb1a2337fdc7ff0b80bbe48cd",
         "RestApiId": {
           "Ref": "ExplicitApi"
         },
@@ -76,7 +76,7 @@
         "CacheClusterEnabled": true,
         "CacheClusterSize": "1.6",
         "DeploymentId": {
-          "Ref": "ExplicitApiDeployment43e01e673d"
+          "Ref": "ExplicitApiDeployment0b29d48047"
         },
         "RestApiId": {
           "Ref": "ExplicitApi"
@@ -244,9 +244,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "ServerlessRestApiDeploymentaa32438b68": {
+    "ServerlessRestApiDeployment56105d1476": {
       "Properties": {
-        "Description": "RestApi deployment id: aa32438b68e05d3771a975585dfbc7b012672b55",
+        "Description": "RestApi deployment id: 56105d1476c2b2ba36a1c595f0c9f323feb5efd9",
         "RestApiId": {
           "Ref": "ServerlessRestApi"
         },
@@ -259,7 +259,7 @@
         "CacheClusterEnabled": true,
         "CacheClusterSize": "1.6",
         "DeploymentId": {
-          "Ref": "ServerlessRestApiDeploymentaa32438b68"
+          "Ref": "ServerlessRestApiDeployment56105d1476"
         },
         "RestApiId": {
           "Ref": "ServerlessRestApi"
diff --git a/tests/translator/output/intrinsic_functions.json b/tests/translator/output/intrinsic_functions.json
index f4eee1a16..34bfb7a0c 100644
--- a/tests/translator/output/intrinsic_functions.json
+++ b/tests/translator/output/intrinsic_functions.json
@@ -243,9 +243,9 @@
       },
       "Type": "AWS::ApiGateway::RestApi"
     },
-    "MyExplicitApiDeployment7145dd00ce": {
+    "MyExplicitApiDeployment011ab6fdbf": {
       "Properties": {
-        "Description": "RestApi deployment id: 7145dd00cea59b4a62b4d7855add490c587f3f62",
+        "Description": "RestApi deployment id: 011ab6fdbfec2a0b749690c4c389b4f44e51a053",
         "RestApiId": {
           "Ref": "MyExplicitApi"
         },
@@ -256,7 +256,7 @@
     "MyExplicitApidevStage": {
       "Properties": {
         "DeploymentId": {
-          "Ref": "MyExplicitApiDeployment7145dd00ce"
+          "Ref": "MyExplicitApiDeployment011ab6fdbf"
         },
         "RestApiId": {
           "Ref": "MyExplicitApi"
diff --git a/tests/translator/test_api_resource.py b/tests/translator/test_api_resource.py
index 0b1340967..51a6c1272 100644
--- a/tests/translator/test_api_resource.py
+++ b/tests/translator/test_api_resource.py
@@ -106,12 +106,91 @@ def translate_and_find_deployment_ids(manifest):
     return deployment_ids
 
 
+@patch("boto3.session.Session.region_name", "ap-southeast-1")
+def translate_with_parameter_values(manifest, extra_parameter_values):
+    """Transform `manifest` with the shared parameter values plus `extra_parameter_values`.
+
+    Region resolution is pinned rather than inherited: without this, partition lookup falls through to the
+    ambient boto3 session and the test fails with NoRegionFound on a machine that has no region configured.
+    """
+    parameter_values = {**get_template_parameter_values(), **extra_parameter_values}
+    with patch("samtranslator.translator.arn_generator._get_region_from_session") as mock_region:
+        mock_region.return_value = "ap-southeast-1"
+        return transform(manifest, parameter_values, mock_policy_loader)
+
+
+def find_deployment_ids(output_fragment):
+    return {
+        key for key, value in output_fragment["Resources"].items() if value["Type"] == "AWS::ApiGateway::Deployment"
+    }
+
+
+def find_stage_variables(output_fragment):
+    return [
+        value["Properties"].get("Variables")
+        for value in output_fragment["Resources"].values()
+        if value["Type"] == "AWS::ApiGateway::Stage"
+    ]
+
+
+def _parameter_driven_stage_variable_manifest():
+    return {
+        "Transform": "AWS::Serverless-2016-10-31",
+        "Parameters": {"EndpointUri": {"Type": "String"}},
+        "Resources": {
+            "ExplicitApi": {
+                "Type": "AWS::Serverless::Api",
+                "Properties": {
+                    "StageName": "prod",
+                    "DefinitionUri": "s3://mybucket/swagger.json?versionId=123",
+                    "Variables": {"EndpointUri": {"Ref": "EndpointUri"}},
+                },
+            }
+        },
+    }
+
+
+@patch("botocore.client.ClientEndpointBridge._check_default_region", mock_get_region)
+def test_redeploy_when_parameter_driven_stage_variable_changes():
+    """A stage variable driven by a template parameter must redeploy when the parameter's value changes.
+
+    Hashing the unresolved {"Ref": "EndpointUri"} yields the same deployment logical id for every parameter
+    value, so the new variable would never be deployed and UpdateStage would point the stage back at the
+    deployment SAM already knows about -- the failure reported in #3703, just via a parameter.
+    """
+    manifest = _parameter_driven_stage_variable_manifest()
+
+    first = find_deployment_ids(translate_with_parameter_values(manifest, {"EndpointUri": "https://one.example.com"}))
+    second = find_deployment_ids(translate_with_parameter_values(manifest, {"EndpointUri": "https://two.example.com"}))
+
+    assert first != second
+
+    # Same parameter value must stay stable, so an unchanged stack does not churn its deployment.
+    third = find_deployment_ids(translate_with_parameter_values(manifest, {"EndpointUri": "https://one.example.com"}))
+    assert first == third
+
+
+@patch("botocore.client.ClientEndpointBridge._check_default_region", mock_get_region)
+def test_parameter_driven_stage_variables_are_not_inlined_into_stage():
+    """Resolving variables for the hash must not leak resolved values into the emitted template.
+
+    resolve_parameter_refs mutates its argument, so hashing a resolved value has to work on a copy; otherwise
+    the AWS::ApiGateway::Stage would emit the parameter's value instead of the customer's Ref.
+    """
+    manifest = _parameter_driven_stage_variable_manifest()
+
+    output_fragment = translate_with_parameter_values(manifest, {"EndpointUri": "https://one.example.com"})
+
+    assert find_stage_variables(output_fragment) == [{"EndpointUri": {"Ref": "EndpointUri"}}]
+
+
 class TestApiGatewayDeploymentResource(TestCase):
     @patch("samtranslator.translator.logical_id_generator.LogicalIdGenerator")
     def test_make_auto_deployable_with_swagger_dict(self, LogicalIdGeneratorMock):
         prefix = "prefix"
         generator_mock = LogicalIdGeneratorMock.return_value
         stage = MagicMock()
+        stage.Variables = None
         id_val = "SomeLogicalId"
         full_hash = "127e3fb91142ab1ddc5f5446adb094442581a90d"
         generator_mock.gen.return_value = id_val
@@ -129,6 +208,52 @@ def test_make_auto_deployable_with_swagger_dict(self, LogicalIdGeneratorMock):
         generator_mock.get_hash.assert_called_once_with(length=40)  # getting full SHA
         stage.update_deployment_ref.assert_called_once_with(id_val)
 
+    @patch("samtranslator.translator.logical_id_generator.LogicalIdGenerator")
+    def test_make_auto_deployable_with_stage_variables(self, LogicalIdGeneratorMock):
+        prefix = "prefix"
+        generator_mock = LogicalIdGeneratorMock.return_value
+        stage = MagicMock()
+        stage.Variables = {"stageVar": "value"}
+        id_val = "SomeLogicalId"
+        full_hash = "127e3fb91142ab1ddc5f5446adb094442581a90d"
+        generator_mock.gen.return_value = id_val
+        generator_mock.get_hash.return_value = full_hash
+
+        swagger = {"a": "b"}
+        deployment = ApiGatewayDeployment(logical_id=prefix)
+        deployment.make_auto_deployable(stage, swagger=swagger)
+
+        LogicalIdGeneratorMock.assert_called_once_with(
+            prefix, str(swagger) + ApiGatewayDeployment._X_HASH_DELIMITER + json.dumps(stage.Variables, sort_keys=True)
+        )
+
+    def test_make_auto_deployable_stage_variables_change_deployment_id(self):
+        swagger = {"a": "b"}
+
+        def deployment_id_for(variables):
+            stage = MagicMock()
+            stage.Variables = variables
+            deployment = ApiGatewayDeployment(logical_id="prefix")
+            deployment.make_auto_deployable(stage, swagger=swagger)
+            return deployment.logical_id
+
+        # Only the stage variables differ, so the deployment must not be reused: reusing it means
+        # the variable change is never deployed, and UpdateStage resets the active deployment.
+        self.assertNotEqual(deployment_id_for({"stageVar": "one"}), deployment_id_for({"stageVar": "two"}))
+
+    def test_make_auto_deployable_without_stage_variables_is_unchanged(self):
+        swagger = {"a": "b"}
+
+        def deployment_id_for(variables):
+            stage = MagicMock()
+            stage.Variables = variables
+            deployment = ApiGatewayDeployment(logical_id="prefix")
+            deployment.make_auto_deployable(stage, swagger=swagger)
+            return deployment.logical_id
+
+        # Templates that set no stage variables must keep their existing deployment logical id.
+        self.assertEqual(deployment_id_for(None), deployment_id_for({}))
+
     @patch("samtranslator.translator.logical_id_generator.LogicalIdGenerator")
     def test_make_auto_deployable_no_swagger(self, LogicalIdGeneratorMock):
         prefix = "prefix"