I have to check the engine warranty, but likely same as in the UK. What's a TuV approval? Insurance issues perhaps? In Sweden the insurance companies had historically not bothered very much with remapped cars (as far as I know), its kind of an semi-legal grey zone.
Audi is likely more concerned with their income, less so for customer who already bought heir car, and the environment (the latter which as I understand is more dependent on driving style) - if they where, they could offer mapping kits as BMW. However, I also understand Audi to some extent, they of course take every opportunity to protect their money and brand - but sometimes is more fruitful to be pragmatic. I myself likely will not dare to put in a remap (unless proven extremely good and clever [i.e. not "seen"] kit) during the warranty period.