It would be interesting to find compelling scenarios where up-to-date information is useful to have, but not so important as to require notification.
At least my phone spends most of its time with the display off, thus live icons are of limited utility. 🙂
An improvement to the multi-tasking API could trigger Apps to update their status when the App becomes visible because of 'display unlocked' or swiping between homescreens. (With some timer to throttle updates to a reasonable interval.)
One example could be a stock-ticker App:
Use Push Notification Service for events that user has subscribed to
New multi-tasking API to update the App icon for a couple of favourite stocks
Yeah, that has always annoyed me. The same with the app store kicking you out whenever you purchase something! Multitasking was implemented for a reason! Download my app in the background!