It's mostly the Rook, of course. While testing other pieces that could be substituted for the Rook, I discovered that the Rook is very much stronger in the endgame than in the opening or middlegame.
The difference is more than half a Pawn, probably two full "points" in many positions.
Of course, the Queen and the Bishops also benefit from the greater freedom of movement they gain as pieces are removed from the board, but not to the same startling degree as the Rook.
Surprisingly enough, a Commoner (a piece that moves like a King but doesn't have to worry about check) is very weak in the opening, reasonably good in the middlegame, and wins outright against a Knight or Bishop in the endgame. (There are no Commoners in FIDE chess, but the value of the Commoner is some guide to the value of the King).
What can you say about the poor Knight? It doesn't get any weaker as the game goes on, it's just that the other pieces get stronger while the Knight stays the same. In fact, it would be reasonable to conclude that the value of the Knight during the opening and middlegame exceeds its conventional value. In fact, that would explain why I have won so many games by allowing my opponent to "win" a Bishop for a Knight in the early part of the game.
It would be good if there were some way to measure piece values at different stages of the game, and given that it would be even better if there were some way of combining the endgame and middlegame values into an overall value.