I don't think the thermostat is your issue. If the thermostat was bad , it would get hot every time you drove it.
Since it takes some time for it to show it self after you fill the system, its most likely loosing coolant slowly over time, either its dripping someplace you have not noticed or its loosing it in the form of steam. The coolant reservoir cap may be bad, and not holding proper pressure, so the coolant will boil at a lower temp than normal causing it to evacuate in the form of steam. I would start by replacing the cap, and looking for any hidden leaks in the system, paying close attention to the rear of the intake, and front of the intake where the thermostat housing is. Also check the oil for the the presence of coolant, and get a test kit to test for the presence of hydrocarbons in the coolant in case you have a slow leaking head gasket.