The valor of King.

Take the pawn as unity.

Pawn = 1

Bishop = 3

Knight = 3

Rook = 5

Queen = 9

The valor of king is absolute, because if you lose it you lose the game, but based in your funcionality, one square in any direction, how many pawns will be?

I read that his fighting value in the endgame is considered about 4 pawns.

Edit:

“In the endgame the king’s fighting value is about four pawns.” - Dan Heisman

If king was a piece like any other then I'd say 1.5 in early, 3 in endgame.

Edit: 2 and 4 actually (4 because he would be able to mate very well in lategame)

Guess it depends on how much "valor" your King has...some fight better than others.

Well...he's definitely an extremely important and valued piece. After all, his fate decides your fate. But, functionally, he's not that good. There are always two kings on a board during a game, they basically cancel each other out. At this point in the game, we're saying it's just kings and a couple of pawns. Naturally, there are probably a couple of pawns on the same file and they will run into each other and be pronounced useless. But if one king takes the initiative and attacks the pawn holding his pawn up, then he most likely will be able to promote, assuming the enemy king isn't closer or promoting his own pawn. So really, in the end-game, kings are mainly guards. They don't do much else. They're like electrons vs electrons or protons vs protons. They can't checkmate each other,  but if they go through a current, they can create electricity. The kings can try to force each other back, but if both kings are doing the sane things, it will be pretty hard to win the game. It'll most likely tie.

'The King is a fighting piece. Use it!' Steinitz.

EVERYTHING IS RELATIVE. But most books agree that on the endgame the king is worth a little bit more than a minor piece AKA 4.