I have a 2013 A3 2.0 tdi and had the same issue. Thankfully I jave an extenced warranty and they replaced the heater matrix and the thermostat. They have told me i may need to top up with coolant as sometimes this will come up as a fault after flushing the system out. I picked the car up yesterday and the heating worked fine and i havent had time to top it up. On the drive this morning the heater worked fine until I turned it off and back on again and its now baclk to blowimg cold.
This is due to the coolant being changed and until I top it up again it will alledgaley not work so the advice ref coolant change seems to be good....will let you know if this doesnt work!
