An executive from a paint company died of hypothermia mountain climbing yesterday.

Hate to be that person, but hyperthermia is too high body temperature.

Hypothermia is below normal body temperature.

In case you tell this again sometime, it might make sense to either change hyperthermia (too hot) to hypothermia (too cold) in the setup or swap out the coroner in the punchline for somebody who would have provided the subject of the joke with advice to put on the second coat before climbing. As it is right now, you've got him dying of excess heat but we're told that he didn't have enough layers.

Nitpick aside, I like the idea you're going for.