Firefox keeps acting "tired" after a while

[Judge]

Now I wonder why it heats up whenever it wants to install updates, then cools down once they're installed...

I'd think that is indicative of your hard drive. Windows updates? Big file transfers, more heat.

Once they're installed then such OS functions and files are more apt to run in resident memory, which generates CPU cycles, but doesn't labor your presumably slow, small cache 5400 rpm SATA hard drive as much.

The important thing is that it does cool down while still being used. Sounds like your fans and heatsinks are likely working as they should. Excess heat in most laptops has always spooked me...if I ever got one I'd be using an SSD and not a mechanical drive. Far less heat. But I'd still be stuck with a miniscule CPU heatsink. Lots of heat there too.

Or to put it another way, the total mass of my computer's heatsink and fan is probably equivalent to nearly a third the size of your laptop. It barely fits in my desktop case!

 

[Xenocity]

Firefox like many web browsers start to slow down and crash when your cache becomes too big.
Also it could be a memory leak.

If you are running a 64-bit OS, use Waterfox (which is a modified version of Firefox) that is optimized for 64-bit machines.
I'm using it and uses all your Firefox files on your system thus you wouldn't know the difference if it wasn't for the program name, logo, and stability.

 

[qwerty]

Probably stating the oobvious here but maybe clogged with adware, do you find your getting a lot of popup ads?

 

[plaidInBlue]

Starting with Firefox 41, the memory footprint will be lower (well, not as deep). Firefox 41 will use less memory when running AdBlock Plus | Nicholas Nethercote
More memory used, more accesses to its content, more page faults. Each page fault generate an interrupt, there's more context switches, the cpu slows down (the clock frequency increases). Translation lookaside buffer - Wikipedia, the free encyclopedia