I run my refrigerator on propane while driving. It is only a tiny burner - 2,200 Btu's. The electric element is around 300 watts - or about 900 Btu's and is not as effective as the gas heater. On a hot day, gas will provide about 75% more cooling than electric.
The 'energy' will come from somewhere. If you have a solar system, then it can come from that source, however it likely will come from the alternator, in increased fuel usage, and that will power the refrigerator. No your fuel mileage will not drop by 2 mpg, but it will increase fuel usage a amount that can hardly be measured. It is only 300 watts per hour, and the alternator would put out about 30 more amp hours per hour to power the inverter (about 450 watts) to power the 300 watt refrigerator. So the engine will use about 1 HP to produce that extra 450 watt load.
Fred.