The coolant temperature sensor is located inside the coolant tank located to the left hand side of the engine. It has two small thin rods inside the tank. If you are having a coolant level warning on your dash but find your water is topped up, then it may be a case that the rods have built up rust and gone faulty. Cheap and easy repair, the coolant bottle does not come apart so its hard to get the rods so best bet is buy a whole new replacement from audi for about £20 and somemore coolant antifreeze water about £8. Three hoses to disconnect, but be careful not to spill the old water and also the coolant temperature connector to come off.
Had to do this on mine a couple of months back.
Audi wanted to charge about £200 to read fault codes, new bottle and labour, absolute joke!!