Falling star killing honest broker and karthus. Does karthus trigger twice?

Yes — but note it's Honest Broker's Deathknell that doubles, not Karthus himself. Karthus's ability is a passive ("Your [Deathknell] effects trigger an additional time"), so it stays active at the moment the trigger is created, even though he's dying in the same event.

Falling Star deals 3 to each unit, killing Karthus and Honest Broker simultaneously. Honest Broker's [Deathknell] then triggers once normally plus once from Karthus, so you get two Gold gear tokens, each entering exhausted.

This differs from the Baited Hook line (FAQ #8551), where Karthus only enters after the Deathknell trigger was already created — there it's a single token. Here he's already on board, so the doubling applies (FAQ #12621; CR 808.1.d.2).