### 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_
for stage in value:
                    patch_operations.append(
                        {
                            "op": "replace",
                            "path": f"/{first_char_to_lower(parameter)}",
                            "value": f'{stage["ApiId"]}:{stage["Stage"]}',
                        }
                    )
                    if "Throttle" in stage:
                        patch_operations.append(
                            {
                                "op": "replace",
                                "path": f'/{first_char_to_lower(parameter)}/{stage["ApiId"]}:{stage["Stage"]}',
                                "value": json.dumps(stage["Throttle"]),
                            }
                        )
            elif isinstance(value, dict):
                for item in value:
                    last_value = value[item]
                    path = f"/{first_char_to_lower(parameter)}/{first_char_to_lower(item)}"
                    patch_operations.append({"op": "replace", "path": path, "value": last_value})
            else:
                patch_operations.append(
                    {"op": "replace", "path": f"/{first_char_to_lower(parameter)}", "value": value}
                )
        client = connect_to().apigateway
        client.update_usage_plan(usagePlanId=usage_plan_id, patchOperations=patch_operations)
class GatewayApiKey(GenericBaseModel):
    @staticmethod
    def cloudformation_type():