API requests are usually encrypted with SSL and protected against unauthorised use with something along the lines of a JWT: https://jwt.io/

Breaking through the SSL might be possible, if the developer doesn't pin certificates, but you don't know the secret used to generate the HMAC signature (blue section of that website), then you can't simulate the API request. And the secret shouldn't be sent over a network connection.

You could probably access the secret with enough work, but it would be a lot of work. You'd have to do it separately for each app. And the developer can change the secret whenever they want. The developer will change the secret at the slightest hint of anything like this being used with their app. And possibly also take additional steps to keep it from being accessed (e.g. store it in the Trusted Platform Module or equivalent on Android/iPhone). Even the CIA can't access that - it's mostly intended for payment processing and protecting data on a stolen phone, but there's nothing stopping a weather app from using it to prevent unauthorised access to their API (weather data is very expensive, and often billed per API request).

Running the real app on a real phone though... basically nothing an app developer can do to stop that.