A cat's sweat glands are concentrated in the paw pads, and almost no sweating happens elsewhere on the body. Wet footprints on a desk in summer, or damp little marks in a carrier before a vet visit, are both common.
The first job of paw-pad sweat is not cooling but grip. When a cat runs, jumps, or turns sharply, slightly damp pads provide better friction.
Stress and excitement also trigger sweating. During a checkup, a car ride, or an encounter with an unfamiliar cat, sympathetic arousal noticeably increases pad sweat, usually easing as the cat calms down.
There are scent glands between the pads as well. Sweat combined with scratching leaves scent on surfaces, which is one of the cat's marking behaviours.
If the pads stay damp for a long time, peel repeatedly, look red and swollen, or come with limping, that is not normal sweating. Check for interdigital dermatitis, a foreign body, or a fungal infection.
Image credit: Sleeping cat paw, by Misterneedlemouse, CC BY 4.0, via Wikimedia Commons: https://commons.wikimedia.org/wiki/File:Sleeping_cat_paw.jpg
