might be a stupid question

the air con will still work even though the dash is not displaying the temp

to set the ambient temperature to show on the display, on VAGCOM go in to number 17, which is instrument cluster coding. go to channel 19. note the original code in case you need to go back for any reason. now change the 4th digit (2nd from the end) to a '1'. this will enable thhe outside temp to be displayed when the ignition is on

To the answer to your orignal post, yes the air con is connected to the ambient temp, it uses the temp as an input signal, example'; if the outside temp goes below 4 degrees the module will not kick the compressor in and therefore the aircon will not work.