I'd previous ran into my horse simply becoming invisible whenever I dismounted it in a desert area or near desert temples. I could still hear it, and after a bit it would re-appear. This was, however, a horse I'd used a nametag on. Incapable of despawning.

Whenever I rode into a similar area, ie. a desert around a temple on an unnamed horse however, dismounting and moving away from the horse it completely despawned. I attempted several times to listen to see if it had simple gone invisible, but no such luck.