It's a push notification. So if your device wasnt connected to the internet, you wouldn't get any notifications. The game server (or any push service they use) will send a notification to your phone when it is supposed to recieve it. It's just like how when you get a message on messenger or a snap on snapchat.This could be due to using more than one device (or using both mobile and PC).
For instance, you use your phone, and when you quit, you have 5 FP on your bar, and the game on the phone knows that in 4h26m your bar will be full. So it schedules internally on the phone a notification to go off in 4h26m telling you your FP bar is full.
You go off and play on another device or on the PC, and spend those 5 FP, but don't log back in to the phone. The phone has no idea you spent those FP and the notification will go off as scheduled, you open the game on your phone and "Whut? My bar isn't full!"