Jumping to a conclusion before diagnosing the problem is a recipe for wasted time and money. The first thing you want to do is make sure the engine is not actually getting hot, then check the sensor, wiring and temperature gauge for proper operation.
3 things, gauge bad, sensor bad or car is overheating. a quick output test with a diag machine will test gauge, comparing coolant temp between cluster and engine will answer the other two. either way, you need a diag tester.
there is the ECT sensor (engine coolant temp.) which would most likely cause the MIL, the engine light, to come on if faulty. and there is also the engine temp. sending unit which may not and could cause the gauge to read incorrectly