Generally, do engines run hotter or cooler if on regular or premium fuel ????

 

Your question about engine heat does not matter. The difference is in the octane rating, the warmer ambient temperature is, the higher octane you need to run because of when it ignites. Denotation is the reason for running premium fuel, not how hot the motor runs. Lower octane in warmer weather can cause the fuel to ignite early. Detonation can cause motor problems.
But, yes i guess lower octane fuel could cause your motor to run hotter.