The fuel injectors may be leaking, due to wear, sticky build-up, or o-ring failure. The warmth of the engine vaporizes the leaking gasoline in the intake manifold, causing an abnormally rich condition upon starting. The longer the car sits, the more excessively rich, requiring more cranking time to clear the intake manifold.
You could check for this ...
I am not familiar with the car model in question, but I have pretty much seen the same behavior in other temperature indicators in cars.
This is an artifact of the indicator design and doesn't mean that the engine overheats.
My first suspicion is with the thermostat. It may be failing "slow" which means it's not opening when it should and it takes the much higher temperature and pressure of the hot coolant to open.
Replace it and I suspect the problem will be solved.