Furnace normally gets it's power from the house fuse box.. Then, at least on an Atwood, it goes to a circuit breaker (looks like an on-off toggle switch) that is on top of the blower housing accessible from OUTSIDE the RV (remove the cover) From there via a MOLEX connector to the control board and from there via a different pin to the thermostat, So if there is no power to the furnace, there is no power to the thermostat.
NOW.. Normally a bad control board would be suspect. but as I recall. not in this case but the switch/breaker is.. Check for power on the control board.
Home was where I park it. but alas the.
2005 Damon Intruder 377 Alas declared a total loss
after a semi "nicked" it. Still have the radios
Kenwood TS-2000, ICOM ID-5100, ID-51A+2, ID-880 REF030C most times