What is the energy cost of an action potential?
There's a few [1, 2, 3] sources that claim it's on the order of $10^8$ ATPs per action potential. The first paper (which is a review that cites the second paper) also has some equations for converting from ATPs to free energy, although that's going to be very context dependent.

If you can't access the third paper, someone has helpfully entered the relevant figures into the bionumbers site.