If I were to guess, you have some application or applications that have integrated multiple mechanisms for delivering push notifications, and their "without google" implementation is of extremely poor quality, like polling for updates every 5 seconds or something like that. After installing google, those applications switched to feeding the notifications through google instead of using their own crap implementations.
Note: I favor push notifications that do NOT use google, but they have to be implemented correctly so that they don't do stupid things like polling servers for updates. A correctly implemented system will have negligible impact on battery life.