Does the cache configured in a custom authorizer within an API Gateway incur any cost?

Hi - Thanks for checking. I haven't seen the pricing on the Authorization Caching on API Gateway pricing. Unless for better performance and faster API execution, you can optionally provision a dedicated cache for each stage of your APIs. After you specify the size of the cache you require, you will be charged the hourly rates for each stage's cache, without any long-term commitments.

Reference : https://aws.amazon.com/api-gateway/pricing/

There is no charge associated to lambda authorizer caching. You will save on costs when you enable authorizer caching since the backend Lambda will not be invoked.