Firefox OS already uses the same model, push notifications will fire up the application, the application can then decide to create a notification, to update the internal cache to reflect the latest changes; it's called the Simple Push API[1]. It can also wake up the application every X minutes to pull the data (Alarms API[2]) and do the same thing.
[1] https://developer.mozilla.org/en-US/docs/WebAPI/Simple_Push
[2] https://developer.mozilla.org/en-US/docs/WebAPI/Alarm