It’s conduction heating at work.
When two objects touch, heat will transfer from a warmer object into the cooler object. Being that your mug of hot water is warmer than your hand, your hand will absorb some of the heat energy, which you register as the mug being warm. The longer you hold on, the more heat energy your hand absorbs, making it feel warmer.
The same thing works in reverse when you touch something cold, you’re not feeling cold, you’re feeling the object you’re touching absorb your body heat.
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