We just went to Disneyworld last month. On the way home, we decided to make the entire trip. 10.5 hours. I turned on the inverter to keep the fridge running and everything went well all the way home.
I'm not 100% sure if mine's live all the time or not. So just in case, I always leave my running lights on when we're towing. That sends power to the camper constantly. Not sure if that makes a difference but like I said, 10.5 hours and full power all the way. We unhooked everything when we got home and the batteries were fully charged enough to work the auto-level and put out the slides.
I'd be hesitant to leave it plugged up for an extended period of time if not running.