The engine coolant fan can keep running after the engine is turned off in any car because the temperature gauge is in the radiator. As long as the radiator is still hot, the fan will keep running until the temperature gauge says it has cooled enough.
If it shows hot on the gauge and it is not hot, there is a gauge or sensor problem.
You probably could A) replace the gauge or B) take it to a dealership and have them look at it
If the gauge is reading hot and the car is running normal temperatures, the water temperature sensor needs replaced.
That depends on what the car is supposed to run at.
Possibly a leak in the coolant reservoir bottle.
Low coolant? Thermostat not opening?
That means the engine is running HOT for some reason. Check the thermostat, Radiator and coolant level. Repair as needed.
check the coolant temp sensor
Sounds like a sticking thermostat. When you say "the engine isn't hot", you mean "the radiator isn't hot", right? Because if the temperature gauge says overheating, your engine ishot
The engine is running hot for some reason. Check thermostat, radiator and coolant level.
Your engine coolant temperature gauge has a sensor ( there is also a engine coolant temperature sensor that is used for the computer / PCM )