1: Does stat run both AC and Furnace? And 2:is it "electronic".
if it is electronic then it needs power.. Normally most Heat/Cool stats get poer from the "heat" side (Furnace) and run on a rechargeable battery during Heating cycles ... That battery may need replacement.
Some use replicable batteries. and again replacement may be needed.
I do not know about yours
Mine Factory the A/C itself (one of 'em) operated the furnace and drew house power.. When that A/C went bad I the new one did not have the ability to control the furnace. So we taped off those leads and I put a mechanical T-Stat on the wall. It did not need batteries.. and since it was "Mercury Free" it did not even need to be level.