My 2003 A6 1.9 TDI ( AVF) gave a lot of problems with the dash indicating over heating ( I know your one has different symptoms ) when it fact, it was not overheating in reality. Changed the temp sensor on the block, new stat, fan belt, timing belt etc. No difference. Would behave itself for awhile, then get up to its old tricks again. Hitting the max within 1 or 2 mins of starting, then just as quick, dropping back to normal. I discovered that the electric fan was not working, and replaced it, but strangely, it was on all the time, so I read here on the forum that if you set the temp to ECON, that stops the fan, so that's what I do now, in case I need the air con, I just switch off the ECON, and bingo, electric fan cuts in. It should not be like this I know, but the fault is in the sensor on the bottom rad water hose. Eventually I got it scanned, and the only fault registered was the Climate Control Module ( CCM, which is located under the carpet at the passenger front seat.) But as it turned out,, the fault is actually in the dash clocks. One nigh while driving home on the motorway, temp went right up, usual flashing gauge on the dash, and loud beeping warning light. I frustration, I thumped the dash panel......and lo and behold,,,,, temp went right down to normal !!! So now any time it starts to rise, I hit the dash, and that brings it back to normal. So in my case, its bad connections or other wiring fault in the dash panel itself. I am not saying that it will solve your problem, but costs nothing to try it anyway !!! Keep us updated with your problem, be Interesting to see what happens in your case.
Al the Best, Jim