If it gets colder when you start driving, that does sound like a knackered thermostat. Very common.
As for the hesitation, does it feel like the boost hits hard and then immediately holds back, then goes again? If so, I would check the O rings in the boost pipes for leaks. A visual clue is oily patches around the clip connections. That is PCV oil being blown out of the connections along with your boost!
If not that, a split DV is another common one, but usually you'll get an underboost fault code with that.