Is there a cheap, safe liquid that evaporates EXOthermically around room temperature?
No such liquid, safe or otherwise, can exist. Evaporation is a strictly endothermic process in all cases.

The change in state from liquid to gas is marked by the individual particles gaining enough translational kinetic energy to overcome the mutual attractions present in the liquid phase to "fly free" in the gas phase. It is logically inconsistent for a substance to increase its internal energy and release energy to the surroundings as heat in the same process.

In order to achieve both evaporation and a release of energy, one would have to find a liquid that reacts to (a) release heat and (b) form gaseous products.  The energy required to move from the liquid to the gas phase is substantial; more than likely the only reactions exothermic enough to provide a net release of heat are combustion reactions.

I somehow doubt dumping a flammable liquid into your thermostat, inserting a wick, and lighting it on fire is a satisfactory solution for you.