Actually, this technology was available to me back in the early 60's. I was working for Schellenger Research Laboratories, on campus at UTEP, and had occasion to fabricate my own precision temperature sensors, and to come up with suitable read-out instruments. The requirement was to precisely measure (to 1/2 degree F accuracy) the temperature at the base of the spark plugs on test engines when running them on a dyno, and on test vehicles.
All that's needed for a sensor are two wires, one iron and the other constantan(with proper insulation and sheath). Just silver-solder the two wires together on the sensor end, and connect the other end to a precision quality dc voltmeter. You'll need to look up the appropriate tables which convert this voltage to temperature. You can even purchase a meter that'll read out in degrees F.
For the subject purpose, a simple circuit with proper components can be used to trigger the cutoff switch at the temperature of your choice. I believe 400 degrees F would work for your RV refrigerator.
Here's a photo of the temperature sensor that I fabricated:
