I have had a similar problem with my air con, no cold air, brought the car to Audi hoping it was not a component fault i.e. compressor failure and it would be a simple re-gas of the system. Audi re-gassed the system and I was told everything was in order, as soon as I drove home there was no cold air, of course not happy. Knew at this point it was definitely a component failure. Having done further research it seems to be regular issue with A3 models that it will fail around 20K mileage or 3 years, whereas normal compressors would generally last 8 years.
Just though I would give you my experience...let us know what your outcome is.