I keep all of my tack in my house and dutifully haul it back and forth. I don't think there's really a problem leaving it in an unheated tack room--did that for years and years. It doesn't "freeze" unless it's wet. But it can be stiffer. And the bits/metal parts of course can get awfully cold.
I like bringing my tack home anyway as too much of mine has "walked away" over the years at various barns. So I'm just in the habit now of hauling.
A good horseman doesn't have to tell anyone...the horse already knows.
cold doesn't hurt leather- it preserves it. It might freeze your "other stuff" like your leather cleaners that you keep in the tack room, but it won't hurt your actual tack.
I switch from metal bits to plastic bits in the winter just to avoid the icy metal problem.