I got a similar problem with my Audi starting about a week ago. The coolant/overheating warning light comes on after 10-15s after I start the engine, cold or hot engine doesn't matter. The coolant level was above minimum and I checked the coolant level sensor: unplugged the connector from the sensor->warning on, i shorted the two pins from the connector just to be sure-> warning still on. In my case, apparently, the Multi-Function Temperature Sensor(MFTS) is the culprit for the warning. This sensor has 3 pins: one for the temp gauge(works fine in my case), one for the ECU and AC control unit to disable AC in case of overheating(AC also works in my case) and a pin that goes to the warning light. I unplugged the connector from this sensor and the warning light didn't came on afterwards, I connected it back and then the warning was on again. I used the car this way for a couple of days and starting yesterday the warning light didn't came on again. Maybe the connectors were oxidized, but I will replace the MFTS this weekend and clean the connector just to be sure. In all this time the engine cooling system worked ok, except the warning light: coolant level above minimum, in stop-and-go traffic if the engine was getting hot the cooling fan started normally, AC worked fine.
I hope it helps.