Seems quite rare on Q3s but often poor hot starting can be caused by an out of range temperature sensor. They still provide data to the engine ECU but the reading could be 20 deg out. As the sensor has not actually failed fault codes are rarely triggered.
Injection timing and fuel mapping parameters vary significantly depending on the air, coolant and fuel temps so the accuracy of any sensor is critical, particularly on start up.
Only way to check is by logging the sensor temp readings from cold. The VCDS scan tool can do this if you know someone with the kit