My Laptop (HP Pentium With An Athlon Processor) Died From Processor Overheating. Which Processors Generate The Least Heat (e.g. Run The Coolest)? Which Laptop Should I Consider For Replacement?

How about an Apple. they are loved by those who have used them and have a cult status because of ease of use and as a secondary benefit they seem to heat up less than other makes.
If you are running your laptop continuously, the fan has to work overtime to keep it cool. I normally lift my lap top with a wedge of books. So the vent where the air goes into the fan is freestanding and not flush against a surface.
Consider this if you are running it for a long time.

Recently there have been all sorts of scares about laptops blowing up or exploding. It almost became an urban legend.
Consider the fact that your laptop now has a rather powerful processor and wasn't designed to run 24 hours a day.
I use mine a lot to write on, and so far haven't had an explosion. I would be devastated.