From THTF P. 306: Chapter 9, "Meanings"

COLD-BACKED HORSE: This is just something people say about a horse that has tightness in his back. After they learn how to teach a horse to feel of them, they aren’t apt to have a cold-backed horse. It’s best not to cinch a horse up tight without moving him around the corral a little bit. Get him to move around the corral (at the walk, trot, and lope and up to the gallop, if there’s a lot of room in there), and if he has some tightness in his body, that should loosen him up. If you knew he had some tightness in his back, you’d check him out on this before working him on the ground or getting on him.