Should apps on iPad Pro work just like on any other iPad iPad Pro resolution size is  2732‑by‑2048. iPad air 2 resolution is 2048‑by‑1536. So if an iOS game was built specifically for all iPad's from Apple, and the app is currently out, does it mean on the new resolution iPad Pro the game app will look like it doesn't fit on the screen?
TL;DR;

It depends on the app. 

More explanation:

Many apps that look fine on all iPhones and iPads are basically the same app. They are written that way to work on all resolutions. (It's a good programming practice not to assume anything about screen resolution of the device the program is run. It's not always possible though).

However, there is the possibility that some apps don't look good on iPad Pro because developers haven't tested their apps on it. For example, in a game, you may view more of the field.