fix(apigateway): arnForExecuteApi fails on tokenized path (#20323)

Fixes #20252.

----

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*By submitting this pull request, I confirm that my contribution is made under the terms of the Apache-2.0 license*

Author: kaizencc

packages/@aws-cdk/aws-apigateway/lib/restapi.ts

@@ -1,7 +1,7 @@
 import * as cloudwatch from '@aws-cdk/aws-cloudwatch';
 import { IVpcEndpoint } from '@aws-cdk/aws-ec2';
 import * as iam from '@aws-cdk/aws-iam';
-import { ArnFormat, CfnOutput, IResource as IResourceBase, Resource, Stack } from '@aws-cdk/core';
+import { ArnFormat, CfnOutput, IResource as IResourceBase, Resource, Stack, Token } from '@aws-cdk/core';
 import { Construct } from 'constructs';
 import { ApiDefinition } from './api-definition';
 import { ApiKey, ApiKeyOptions, IApiKey } from './api-key';
@@ -368,7 +368,7 @@ export abstract class RestApiBase extends Resource implements IRestApi {
   }
 
   public arnForExecuteApi(method: string = '*', path: string = '/*', stage: string = '*') {
-    if (!path.startsWith('/')) {
+    if (!Token.isUnresolved(path) && !path.startsWith('/')) {
       throw new Error(`"path" must begin with a "/": '${path}'`);
     }
 

packages/@aws-cdk/aws-apigateway/test/restapi.test.ts

@@ -1,7 +1,7 @@
 import { Template } from '@aws-cdk/assertions';
 import { GatewayVpcEndpoint } from '@aws-cdk/aws-ec2';
 import { testDeprecated } from '@aws-cdk/cdk-build-tools';
-import { App, CfnElement, CfnResource, Stack } from '@aws-cdk/core';
+import { App, CfnElement, CfnResource, Lazy, Stack } from '@aws-cdk/core';
 import * as apigw from '../lib';
 
 describe('restapi', () => {
@@ -424,6 +424,16 @@ describe('restapi', () => {
     expect(() => api.arnForExecuteApi('method', 'hey-path', 'stage')).toThrow(/"path" must begin with a "\/": 'hey-path'/);
   });
 
+  test('"executeApiArn" path can be a token', () => {
+    // GIVEN
+    const stack = new Stack();
+    const api = new apigw.RestApi(stack, 'api');
+    api.root.addMethod('GET');
+
+    // THEN
+    expect(() => api.arnForExecuteApi('method', Lazy.string(({ produce: () => 'path' })), 'stage')).not.toThrow();
+  });
+
   test('"executeApiArn" will convert ANY to "*"', () => {
     // GIVEN
     const stack = new Stack();