Thanks everyone

I've tried all four, or maybe that's three, of those options.
A cooling pad isn't enough with bottom section of the case on.
Removing the animal hair was quite satisfying and helped a little but not enough.
Careful vacuuming also helped get gunk out but ....
I haven't yet tried removing the bottom of the case and using a fan pad and I expect this might work quite well. I calculate that blasting air over the outside of the "shroud" that carries air internally from outside the computer via its internal fans to waft over the processor, might greatly help with heat dissipation but the problem is happening at a relatively low 27C and there's a fair chance that the summer temp. here will go above 35C and occasionally above 40C. I suspect that even with a fan pad and the bottom off the case, there'd still be problems. As I said above, this problem is happening even when the processor isn't working hard.