Yep, had this exact problem at the start of the cold weather this autumn. Turning it on the key for 15-20 secs, then the occasional warning beep!
Coolant temp sensor. 20quid from audi. Think it must show a sort or open circuit, making the ECU think there is no coolant in whatsoever, then it changes value to show something different when some water is flowing past it.
It's just a thermistor with a plug for a loom on the other end.
Piece of wee-wee to change too!!