Can I respond to the temporary to return Karthus to the hand before the Deathknell is applied?

Yes, you can respond to the Temporary triggered ability.

According to RiftJudge FAQ #6422 and FAQ #9384, the Temporary keyword is a triggered ability that creates a chain. Because a chain is created, the game enters a Closed State, allowing players to play [REACTION] speed cards in response to the Temporary trigger.

Sequence of Events:

1. Trigger: The Temporary ability triggers at the start of your Beginning Phase.
2. Chain Creation: The Temporary trigger is added to the chain as a pending item. The state is now Closed.
3. Reaction: You can now play a [REACTION] card (such as a spell that recalls a unit to hand) to target Karthus, Eternal while he is still on the battlefield.
4. Resolution:
   • Your reaction resolves first (due to LIFO - Last-In, First-Out), returning Karthus, Eternal to your hand.
   • The Temporary trigger then resolves, which instructs you to kill the unit (assuming it is still there or if the instruction is applicable).

Note on Passive Abilities: If Karthus, Eternal is returned to your hand before the Temporary effect resolves, his passive ability ("Your [Deathknell] effects trigger an additional time") will no longer be active on the board. Therefore, if other units you control with Deathknell are also triggered to die during this sequence, they will not benefit from his passive bonus because he is no longer on the battlefield at the time their Deathknell triggers are processed.