Are they always lost when the enhancement goes away, like a raging barbarian's hit points? Or does it work like temporary hit points and they're lost first (so if the only damage your item has taken was soaked by the temp. bonus hp, it ends up unscathed)?
For any source of temp. enhancement bonus, unless it actually makes a difference. I was thinking of a Magus's arcane pool, where those hp go away in a minute or less, and the answer to this question could thus matter very quickly in-game.
The game doesn't call out that items get temporary hp, so mechanically it would work like a barbarian in your post by default.
I would have figured it would default to how temp hp work, since that is the more common way temporary effects are handled, and Barbarian's Rage is fairly unique.
Why would they default to how temporary hp works? The enhancement increases, the stats derived from it would change. Later when the enhancement changes (decreases) those stats would revert, the damage still remains and would be applied to the new stats. It is how it works with Bear's Endurance (and anything derived from such) as well as rage. It isn't "unique" at all.