It will depend on whether you have +12v available at the thermostat. I don't believe the AR7815 needs it. A digital thermostat does.
If you don't have +12v available, your best choice is a battery-powered thermostat. There are dozens of residential ones to choose from.
I replaced mine with the Coleman Mach RV Comfort.PHP, but I also wanted the separate gas/electric heat settings, since I also installed the electric heat kit in my AC unit. They have a model without that feature.
If you're not comfortable sorting out all of the wires, I would call any HVAC repair company and have them do it. They install similar units daily. There is nothing special about RV thermostat circuitry (other than using 12v instead of 24v).
Whatever you install, it will be far more stable and comfortable than the one you have.