CloudFormation doesn't deploy to API gateway stages on update

Seems like there is no way to easily create a new Deployment whenever one of your Cloudformation Resources changes.

One way to work around that would be to use a Lambda-backed Custom Resource (see http://docs.aws.amazon.com/AWSCloudFormation/latest/UserGuide/template-custom-resources.html).

The Lambda should create the new Deployment, only if one of your Resources has been updated. To determine if one of your Resources has been updated,
you will probably have to implement custom logic around this API call: http://docs.aws.amazon.com/AWSCloudFormation/latest/APIReference/API_DescribeStackEvents.html

In order to trigger updates on your Custom Resource, I suggest you supply a Cloudformation Parameter that will be used to force an update of your Custom Resource (e.g. the current time, or a version number).

Note that you will have to add a DependsOn clause to your Custom Resource that will include all Resources relevant to your API. Otherwise, your deployment might be created before all your API Resources are updated.

Hope this helps.

When your template specifies a deployment, CloudFormation will create that deployment only if it doesn't already exist. When you attempt to run it again, it observes that the deployment still exists so it won't recreate it, thus no deployment. You need a new resource id for the deployment so that it will create a new deployment. Read this for more information: https://currentlyunnamed-theclassic.blogspot.com/2018/12/mastering-cloudformation-for-api.html

CloudFormation in Amazon's words is:

AWS CloudFormation takes care of provisioning and configuring those resources for you http://docs.aws.amazon.com/AWSCloudFormation/latest/UserGuide/Welcome.html

Redeployment of APIs is not a provisioning task... It is a promotion activity which is part of a stage in your software release process.

AWS CodePipeline is a continuous delivery service you can use to model, visualize, and automate the steps required to release your software. http://docs.aws.amazon.com/codepipeline/latest/userguide/welcome.html

CodePipeline also supports execution of Lambda functions from Actions in the pipeline. So, as advised before, create a Lambda function to deploy your API but call it from Codepipeline instead of CloudFormation.

Consult this page for details: http://docs.aws.amazon.com/codepipeline/latest/userguide/actions-invoke-lambda-function.html