There must be some part of the engine, radiator, pipes, heater or hoses that are getting hot. Its impossible for an engine to run without generating heat.
When you start from cold the thermostat is shut so as to restrict the circulation around the block and heater.
This does two things, lets the engine warm up to its operating temperature quickly and get the heater going to keep you comfortable.
If the thermostat is stuck open this wont happen quickly, as the radiator is cooling it quicker than it can get warm.
You wont notice this in the summer. but as they changed the thermostat thats not your problem.
If the circulation is not taking place because the pump isn't working, the heat will build up in the block and only get to the other parts of the system by direct heat transfer. it could take a long time to reach the heater depending on the design and layout of the cooling system
You need to run the car for say 10 mins, and put your finger on all the points that comprise the system to see which are hot and which are not. in that way you should be able to work out if the pump is working or not.
I must say, I am surprised the garage didn't know how to do this simple check.
