i had the same problem and it was definitely the thermostat. it was cracked and a piece was broken off on one side so it could not close. i went out and bought a new one for like 12 bucks or so and installed it in about 10 minutes. nothing to it, definitely anybody can achieve this minor repair.

once you put it in you will get HOT air blowing into the car again

Good Luck

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Audi TT