Endgame position

This is from my game against IM Esben Lund played last year. I am white and it's my move. Short on time and defending for a long time we ended up in this position and I found nothing better than 81.Rf8+ and I resigned after 81...Bf5. The thing is black is threatening ..Bf3+ and then ..Bg4 to close the g-file and then promote next move and my king can't help to stop the g-pawn.

Can you find an improvement or was this the natural result of this game?