Q: Temperature Gauge

With most any other vehicle, your diagnosis would be dead on, congratulations! In your case, the problem is most likely the thermostat. With this Audi engine, the Audi engineers were more clever, there is only a single coolant temperature sensor, which the engine management system uses for fuel mixture, AND drives the gauge on your dash. When the engine computer doesn’t see engine temperature come up as quickly as the stored profile specifies, it makes the dash temp gauge dysfunctional as you note. If it happens more than once, the Check Engine light is turned on with a coolant system problem code. Replace the thermostat, clear the codes, and your gauge will function as expected once more. I would recommend a Thermostat Replacement service from YourMechanic.com to confirm this diagnosis, and make this repair. YourMechanic.com will dispatch a mobile, professional mechanic to your location for this repair. Contact YourMechanic.com to request this service and schedule an appointment.