We have traveled many times in cold weather, December, January, and March with our previous travel trailer. The refrigerator was petty much a standard gas-electric RV refrigerator with the freezer compartment on top with it's own separate door.
First, the refrigerator in all our previous campers are never turned off. We leave our campers plugged in 24x7x365 and the refrigerators have always been set on Auto, so if electric power goes out they would run on propane.
We never had problems running the refrigerator while traveling. It would flip over to gas once unplugged from shore power and just keep things nice and cold. I never tried to analyze it, I just did it.
I do know, that if the inside of the trailer gets really cold ... like 20 degrees below zero ... which did happen a few times over the years ... the refrigerator just quits running. Not because there is anything wrong, but because the thermostat in the refrigerator is reading temperatures that have penetrated the inside that colder than the refrigerator settings.
If the ambient temperature is zero degrees and your refrigerator is set for 38 degrees, well, it's just simply not going to run, because it's already cold.
This was fine for the refrigerator section. But for the freezer section, the ambient temperature had to penetrate below the refrigerator setting before it stopped working. In other words, it had to be really, really, really cold a long time for the outside temperature to penetrate to the inside of the camper, and then the temperature of the trailer to penetrate to the inside of the refrigerator before the refrigerator thought it was already cold.
So, basically what I'm saying is, if you are taking any frozen foods or cold foods with you and you are leaving from a snow, very cold location, just turn your refrigerator on a day before heading out. If you have heat on inside the camper, the refrigerator will run as normal.
Load the refrigerator once is good and cold, unplug shore power, let it flip over to gas and hit the road. Things WILL stay cold and frozen. If the ambient temperature inside the camper hovers freezing or below, the refrigerator and freezer simply won't run very much, but it will stay cold and frozen. Once the ambient temperature inside the camper reaches above the temperature settings on the refrigerator, then it will kick on. Either way ... your ice cream will not melt ... been there ... done that.
Now, our current camper has an all electric residential refrigerator in it. The camper is closed up right now, but still plugged in. The outside temperature is 20 degrees or less at night now. The inside of the camper is COLD! I've checked the temperature in the refrigerator, both the freezer and the refrigerator section. It's staying right on 38 degrees and negative 2.
I'm curious what it will do when temps inside the camper get down to zero (which will probably happen soon enough here in Indiana).
Long story short, turn on the refrigerator on an Auto setting (gas-electric), or just electric if it's residential. Let it cool down. Load it, and take off! No worries.