Replacing the mechanical thermostat with a digital programmable one is worth it! I replaced mine and it has kept the temperatures very stable. I also like that it turns way down at night to reduce battery & propane use (we have good sleeping bags), then comes back up in the morning BEFORE I need to get up. Here's a link to how I did my replacement:
Thermostat Replacement.
One thing I haven't seen mentioned yet is that the furnace blower needs to run at a fairly high speed (i.e. needs higher voltage) before the safety interlocks allow the gas to turn on and ignite. The furnace not igniting is usually the first sign of a weak battery. It will blow cold air for a little while then shut off. My experience is that it does this a littel before the battery hits 50% SOC. So when the furnace doesn't light, it's time to recharge; unless it's 3AM AND we're in a campground. If boondocking, the generator fires up whenever I need it to!