When a seal is given temporary, can you tap it in response to the temporary trigger and keep the power during your turn?

You can tap it in response — but the power won't survive to your Main Phase, so you can't keep it.

Seal abilities have Reaction (exhaust: [Reaction] — Add power), so they're legally timed during the Closed State while the Temporary trigger sits on the chain in your Beginning Phase. You tap, the power finalizes immediately, and then the trigger resolves and kills the seal. So yes, you get the 1 power — but your Rune Pool empties at the start of your Main Phase (Rule 316.3), and the Beginning Phase comes before Channel and Draw. The power is wiped long before you'd have anything to spend it on.

The only exception: if some effect during the Beginning/Channel/Draw phases itself has a power cost, you could use it there. Otherwise the tap is wasted — the seal dies either way.