What happens to my horse if I ride another horse?

Frost only becomes your horse if you persuade Louis Letrush to let you keep him or kill after you talk to him after you have stolen Frost from the Black-Briar Manor. If you just ride off without completing the quest Frost will just return to where you first stole him (Black-Briar Manor).

If you own him and you were to mount another horse (at least another horse you own) he will remain at the location you left him so far as I can tell. I use Shadowmere (who seems to do the same thing) and I just parked Frost at a stable so at least he would be safe. He has stayed there since.

Also any horse you bought from a stable will return to that stable when you begin using another horse. When I first got Shadowmere the horse I purchased from Whiterun began to walk away and when I next went back to the Whiterun stables he was there safe and sound.

I've got some experience with owning multiple horses, basically:

You can own as much horses as available, but only one owned horse is marked as active, which is the horse you last used.

Any non-active horse will automatically return to its stable when you fasttravel on your horse, or directly into a city.

The active horse will always travel with you, or to the nearest stable if you go directly into a city.

This can cause 2 owned horses to be at a single stable, your active horse, and the horse that belongs to that stable.

For the sake of completeness, If you ended up with frost at the end of the quest, it is considered an owned horse in all regards.

From what I've seen it'll go to it's original home.

Say you own (bought) a horse, you find another one (either steal or find one in the while that is just free) you get on it, the new horse will NOT be yours, it'll always return to where you first found it, and at the same time your own horse will go back to the stables you bought it from. You can keep fast traveling and neither horse will be with you till you go to your own horse and mount it again.

With Frost this could work differently, but why not save the game and try out a few things?