Most every new heater I've seen in the last decade had a thermostat that only went down to about 35 or 40 degrees. It cannot be turned down to "off" that can only be done by means of the switch. If it has an on / off switch separate from the thermostat it usually means the unit is designed to prevent freezing.
They usually call it "frost guard" or something to that effect.
If your heater has that function adding a second thermostat into the mix only increases the chances of failure.
Those thermo cubes are designed to turn on & off devices like heat lamps or other heating devices without thermostats.