yes, I keep mine on.
Many now have residential type electric only fridges, and those would be much more limited by battery capacity, but the typical gas or electric fridges can run a LP.
Mine lives in auto, LP is lit, unless I'm running the generator, in which case it switches over.
Many say that you should turn it off when pulling into a gas station, although as I understand it most folks don't.
If you want to understand much more about how these things work, and the reasoning why you should only run them when level, check out this site...
https://www.arprv.com/
He has lots of information on them posted which I think is interesting even if you have no interest in his contoller
In this regard, you might learn that prolonged uphills and downhills may point you to turning the thing off...but I think most folks believe that the motion of driving precludes this.
Me, I have one of their controllers installed. One of the first things I did when we bought the MH. I sleep better hoping at least that the fridge is protected