If you use Ornn Fire Below the Mountain ability on turn 1 does the added power stay on your next turns?

No, the power added by Ornn's "Fire Below the Mountain" ability does not stay in your rune pool for your next turn.

According to Rule 163 and Rule 163.1:

"Every player's Rune Pool empties at the end of each player's draw phase and the end of each player's turn. Any unspent Energy or Power are lost."

Sequence of Events:

1. Activation: You activate the ability on Ornn, "Fire Below the Mountain."
2. Resolution: The ability adds the specified power to your rune pool.
3. Usage: You may spend that power during your turn to play gear or use gear abilities, as restricted by the card's text.
4. End of Turn: When your turn ends, your rune pool empties, and any unspent power—including any power added by Ornn's ability—is lost.

Because the rune pool is cleared at the end of every turn, you cannot accumulate power from this ability to use on subsequent turns.

Note: I could not fully verify the activation method of this ability against the retrieved rules.