A common problem on that engine is the coolant temp sensor.
They go down for fun.
I would not think that the thermostat is at fault , you dont get many problems with them , but the coolant temo sensor will read how hot the engine is and let the temp guage know and it will alos let the engine run rich as it will adjust the fuelling to suit.
I believe that it is mounted in the thermostat housing though , and I am sorry I dont know where that is
Had a problem for about three weeks - for the first few days it got to 70 and sat, most of the time since then it has not really got more than a faction above 50. Fuel economy has gone through the floor.
It's a 2002 2.0
Defo sounds like the thermostat. The stat should be closed when the water temp is cold thus stopping water flowing through the radiator. Sounds like yours is opening too early so its cooling the water too much and not letting your engine reach its normal operating temp.
Forgot to say that I changed mine on my 1.8T (AGU) Golf in about 1 hour and cost £12 from GSF. Not sure if it's similar on the Audi. I've got the 163bhp 1.8T in my A4 but haven't looked if it's the same.