if my opponent uses kharox to play a unit from my trash. and i use temporal breach on it. what happens? who gets it? why?

The unit is banished for good — it stays in banishment, and neither you nor your opponent gets it.

You remain the unit's owner even though your opponent played it from your trash via Kharox — ownership never changes (Rule 127.1). When Temporal Breach resolves, it banishes the unit and then instructs its owner (you) to play it to the same location — which is your opponent's base or a battlefield they control. You can't legally play a unit to a location you don't control (Rule 355.2.a), so that instruction is impossible and is ignored (Rule 359.3.e.6). The banishment stands.

This is the same principle as the Hostile Takeover + Temporal Breach ruling (FAQ #11767): the card isn't "yours to play back" in any meaningful sense — the replay instruction just can't be followed, so the unit remains removed from the game.