I leave my game open when my computer is on, which is quite a bit. If I'm going away from it for a while, I'll exit the game so the notifications don't come up while I'm not there. I'm not sure if they stop the action.

Sometimes I think their energy levels do increase if you exit the game, but I think that's it more a night thing - so if it's on during their day time I don't think it makes much difference. They seem to keep pretty well rested with naps.