### Is there an existing issue for this?

- [X] I have searched the existing issues

### Current Behavior

APiGateway Model that being created by CDK project stores the JSON schema with a single quotes instead of double quotes which leading to an error in parsing this json object. 

you can refere to this article to see the steps 

https://dev.to/dvddpl/reduce-lambda-invocations-and-code-boilerplate-by-adding-a-json-schema-validator-to-your-api-gateway-15af

and it works fine on AWS account 

also you can look on the attached image how localstack stores the JSON object with a single quotes, I tried to edit it manually through UI but it return it back with single quotes

![Screen Shot 2022-10-18 at 4 28 08 PM](https://user-images.githubusercontent.com/106184681/196769273-0834865a-2fb3-4efc-b3a9-9da6b2ad511d.png)

### Expected Behavior

store the json schema in correct JSON format

### How are you starting LocalStack?

With the `localstack` script

### Steps To Reproduce

#### How are you starting localstack (e.g., `bin/localstack` command, arguments, or `docker-compose.yml`)

    localstack start

#### Client commands (e.g., AWS SDK code snippet, or sequence of "awslocal" commands)

    cdklocal deploy


### Environment

```markdown
- OS:OSX
- LocalStack: 1.0.4
```


### Anything else?

_No response_
region_name = invocation_context.region_name or aws_stack.get_region()
    client = aws_stack.connect_to_service("apigateway", region_name=region_name)
    authorizers = client.get_authorizers(restApiId=invocation_context.api_id, limit=100).get(
        "items", []
    )
    for authorizer in authorizers:
        run_authorizer(invocation_context, authorizer)
def validate_api_key(api_key: str, invocation_context: ApiInvocationContext):
    usage_plan_ids = []
    client = connect_to(
        aws_access_key_id=invocation_context.account_id, region_name=invocation_context.region_name
    ).apigateway
    usage_plans = client.get_usage_plans()
    for item in usage_plans.get("items", []):
        api_stages = item.get("apiStages", [])
        for api_stage in api_stages:
            if (
                api_stage.get("stage") == invocation_context.stage
                and api_stage.get("apiId") == invocation_context.api_id
            ):
                usage_plan_ids.append(item.get("id"))
    for usage_plan_id in usage_plan_ids:
        usage_plan_keys = client.get_usage_plan_keys(usagePlanId=usage_plan_id)
        for key in usage_plan_keys.get("items", []):
            if key.get("value") == api_key:
                
                api_key = client.get_api_key(apiKey=key.get("id"))
                return api_key.get("enabled") in ("true", True)
    return False
def is_api_key_valid(invocation_context: ApiInvocationContext) -> bool: