If running on electric, it can take 8 - 24 hours to get from 85F to 40F. It should be cold by morning, enough to put in your cold stuff from the home refrigerator anyway. Running on gas, you get about 2,200 Btu's input to the flame, the boiler will see about 80% of that, or say 1,800 Btu's.
The 120 volt 300 watt electric element if running at 120 volts will produce around 900 Btu's, at 110 volts will have less amperage,and less watts, so will be closer to 800 Btu's. The 12 volt element if you have one will be only a nominal amount of power, say 100 watts or 340 Btu's - and only for a very short time until the battery is dead.
So change over to gas on a hot summer day, it will not only make the RV's power cord carry 3 less amps, it will also provide twice the cooling capacity.
So the answer is yes the refrigerator should cycle off once the temperature is below 40F in the refrigerator, and below abount 0F in the frezer section. However on a hot day, running on electric, it might run 20 - 24 hours before cycling off, and might go well above 46F in the refrigerator section during a typical 100 - 105F day. The freezer can reach 20F on a warm day too. Yet running on gas, the burnger will cycle on and off as needed to keep it cold.
My guess is you heard the fan running near the battery charger / converter. That fan might run for about 1 - 2 hours after plugging in, while giving the battery a good recharge.
I would also suggest checking the battery water level, and refilling it with water if needed, - distilled water only!
Fred.