Amazon Cognito was not designed to secure developer built APIs and I would caution you from using only the Amazon Cognito token to secure your API.
That being said, the vended Amazon Cognito token is a normal JWT signed using asymmetric encryption. This thread on the AWS forums has some example code in C# that another customer was able to use to verify the token.
Update 2015-07-09 AWS has announced Amazon API Gateway. Using API Gateway you can build a REST interface to your existing API (or to AWS Lamdba functions) secured with credentials retrieved via an Amazon Cognito authflow. See this blog post for additional announcement details.