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I agree with Budward that TVs aren't always set up from the factory (or correctly set up) to supply the electric needed to run the fridge, and recharge the battery. Both can be very large loads.
I also usually use LP on the road, to prevent any problems. That leaves all of the supplied amps for the battery, if it is low. It is also one less step to worry about when boondocking - when I get there I'm already in "boondocking mode".
Good tip on the thermometer. That is how I discovered our fridge wasn't working properly on day 2 of a 25 day trip, several hours after we got on the road that 2nd day. Glad I packed a cooler as a backup, although it was pain.