How to heat a cooler without power.?

depends on the exact setup - what's inside the cooler? There are coal-burning handwarmers, oil-burning handwarmers amd, at a pinch, tea lights if the contents of that cooler don't have to worry about CO/CO2 poisoning. You<'ll just have to try for how long these will be burning (in the closed cooler) or how big ventilation holes your setup could tolerate under which external conditions.

60, to 80 degrees, is a perfect temperature for your cat, or dog. Put a pillow inside, and let the cat sleep in there overnight. Prop the lid open, of course.

Well, if you want to use a solar setup, you'd just buy a kit from somewhere, they sell them in camping stores.

I was going to suggest using the "blue ice" kind of ice packs, but warmed instead of cooled. Same idea as the warm stone that Bob suggested.

Chemical handwarmers.