A 2011 A3 was taken into my garage last month. The car owner said it came up with an engine check light the other day. One day after the light was on, he had a fairly long run and ran the heating full but it never blew very hot. I start the car; it ran ok but I noticed the temp gauge reading was approaching approx. 70deg even from cold. I'm assumed it could be a short. I plugged VAS 5054A scanner in its OBD2 port and got two codes 17513 and P1105, which indicates O2 sensor heating circuit Bank 1 - sensor 2 short to B+.
O2 sensor heating circuit bank 1 relates to the pre-cat sensor, bank 2 relates to the post-cat sensor. I checked both the sensors and read the live data with VAS 5054A Bluetooth, the sensors and the circuits are ok. I erased the two error codes, and was able to turn off the light. However, the temp gauge kept going up.
It seemed the temp gauge was faulty. I replaced its stat and car runs at 90deg according to the live data but the gauge reads 110deg. Once it hits 94deg (114deg) the fans kick in so it's doing the job just not reading right. I replaced the temp sensor and the issue was the n fixed.