When I run CloudFormation deploy using a template with API Gateway resources, the first time I run it, it creates and deploys to stages. The subsequent times I
I may be late, but here are the options which which you do a redeployment if a API resources changes, may be helpful to people who still looking for options -
Try AutoDeploy to true. If you are using V2 version of deployment. Note that you need to have APIGW created through V2. V1 and V2 are not compatible to each other. https://docs.aws.amazon.com/AWSCloudFormation/latest/UserGuide/aws-resource-apigatewayv2-stage.html#cfn-apigatewayv2-stage-autodeploy
Lambda backed custom resource, Lambda inturn call createDeployment API - https://docs.aws.amazon.com/AWSCloudFormation/latest/UserGuide/template-custom-resources.html
CodePipeline that has an action that calls a Lambda Function much like the Custom Resource would - https://docs.aws.amazon.com/codepipeline/latest/userguide/actions-invoke-lambda-function.html
SAM(Serverless Application Model) follows a similar syntax to CloudFormation which simplifies the resource creation into abstractions and uses those to build and deploy a normal CloudFormation template. https://docs.aws.amazon.com/serverless-application-model/latest/developerguide/what-is-sam.html
If you are using any abstraction layer to cloudformation like Sceptre, you can have a hook to call createDeployment after any-update to the resource https://sceptre.cloudreach.com/2.3.0/docs/hooks.html
I gone with third option since I kept using Sceptre for Cloudformation deployment. Implementing hooks in sceptre is easy as well.