fix: api key is associated to the default stage

lib/apiIntegration.ts

@@ -1,9 +1,12 @@
 import {
+ ApiKey,
  ApiKeySourceType,
  AwsIntegration,
+ Deployment,
  EndpointType,
  JsonSchemaType,
  RestApi,
+ Stage,
 } from "aws-cdk-lib/aws-apigateway";
 import { REGION } from "./common";
 import {
@@ -25,6 +28,7 @@ export const getApiIntegration = (
  description: "Integration API to SFN",
  endpointTypes: [EndpointType.REGIONAL],
  apiKeySourceType: ApiKeySourceType.HEADER,
+ deploy: true,
  });
 
  const plan = integrationApi.addUsagePlan("UsagePlan", {
@@ -35,8 +39,9 @@ export const getApiIntegration = (
  },
  });
 
- const key = integrationApi.addApiKey("ApiKey");
- plan.addApiKey(key);
+ const apiKey = integrationApi.addApiKey("ApiKey");
+ plan.addApiKey(apiKey);
+ plan.addApiStage({ stage: integrationApi.deploymentStage });
 
  const apiGatewayStartUpdateIpSFNRole = new Role(
  scope,
@@ -131,6 +136,6 @@ export const getApiIntegration = (
  });
 
  new CfnOutput(scope, "IntegrationApiKey", {
- value: key.keyId,
+ value: apiKey.keyId,
  });
 };