Currently the built in authorization provided by the Cloud Run Invoker IAM role will only validate a JWT token with an aud (audience) claim that is set to the domain issued by the Cloud run service (ending in
run.app) and not custom domains. You can still set the URL to the custom domain, but you will need to set the audience explicitly when configuring the push subscription (it defaults to the same as the URL if not set explicitly).
For fully automated system push subscriptions, what is the advantage to you of using the custom domain?
-Preston