If falling star targets both my opponent’s karthus and honest broker, does honest broker deathknell go off twice?

Yes — you (as the caster) kill them, but your opponent controls both units, so Karthus's passive doubles Honest Broker's Deathknell and your opponent gets two Gold gear tokens (each entering exhausted).

Falling Star deals its damage in one resolution, so Karthus and Honest Broker die simultaneously in the same cleanup. Karthus's "Your [Deathknell] effects trigger an additional time" is a passive, not a triggered ability, so it's still active at the moment Honest Broker's Deathknell trigger is created — it doesn't matter that Karthus is dying in that same event. Both are controlled by the same player, so it counts as one of "your" Deathknell effects (FAQ #12621; Rule 808.1.d.2).

The trap is ownership of the doubling: if the two units belonged to different players, the Karthus player's passive would apply only to their own Deathknell effects, and only the Karthus controller's units would get the extra trigger.