The plan is to march the white king towards the pawn: e.g. Kc3-d2-e3-f4-g5-g6-g7
However black then checks the white king perpetually.
The problem is that pawn h7 is too far advanced. If the pawn were on h6, then the white king could hide at h7 and then Rh8 can move to g8 and g6 to allow Kg7 and promotion h8=Q.
Tips on Practical Rook Endgames

So this is essentially a Vancura position with extra pawns?
Yes, but the passed pawn is already past the 6th rank, so it is a draw.

In this particular example Black has no structural problems on the queenside, so with correct play it is a draw even with the pawn at h6.

In this particular example Black has no structural problems on the queenside, so with correct play it is a draw even with the pawn at h6.
Really? Why can't White go for the plan tygxc said?
#11
It does not work here. White can queen the pawn and thus win the black rook, but in the mean time the black rook will have captured the white queen's side pawns that the king has abandoned and then that leaves a drawn R vs. pawns endgame.

Black just shuffles his rook back and forth. This isn’t really hard; if white advances his pawns then black just trades them off.

In this particular example Black has no structural problems on the queenside, so with correct play it is a draw even with the pawn at h6.
Really? Why can't White go for the plan tygxc said?
Often, even with the pawn on h6, it's drawn because, although the king can go over and win the rook for the pawn, that leaves the defending side with enough time to win some queenside pawns and get into a drawn pawns-vs-rook ending. In this case, it's actually a hinderence to White that it's such a far outside pawn, since it's White's king that has to go on the long journey to the pawn. (Usually, of course, it's the defender who has to worry about the remoteness of an outside passer.) So winning chances are actually higher if if it's a g-pawn than an h-pawn in a situation like this.
I believe there are some highly advanced, technical ideas for playing for a win here (or at least with the pawn on the 6th, rather than 7th rank), that were found in the last decade or so with engine assistance. Dvoretsky covers them, if memory serves. But I don't know how practical they are if you don't have a really deep grasp of the endgame in question.

There is a way to play for a win
Push all the pawns and hope for a pawn breakthrough
Black can definitely draw but it isn't brainless

yeah after 1. a5 blacks only way to draw is 1...a6 2. bxa6+ Ka7

In this particular example Black has no structural problems on the queenside, so with correct play it is a draw even with the pawn at h6.
Really? Why can't White go for the plan tygxc said?
Often, even with the pawn on h6, it's drawn because, although the king can go over and win the rook for the pawn, that leaves the defending side with enough time to win some queenside pawns and get into a drawn pawns-vs-rook ending. In this case, it's actually a hinderence to White that it's such a far outside pawn, since it's White's king that has to go on the long journey to the pawn. (Usually, of course, it's the defender who has to worry about the remoteness of an outside passer.) So winning chances are actually higher if if it's a g-pawn than an h-pawn in a situation like this.
I believe there are some highly advanced, technical ideas for playing for a win here (or at least with the pawn on the 6th, rather than 7th rank), that were found in the last decade or so with engine assistance. Dvoretsky covers them, if memory serves. But I don't know how practical they are if you don't have a really deep grasp of the endgame in question.
I just took a look at it, and you're right; the white takes too long and black can win the pawns plus get back on the a file on time

In this particular example Black has no structural problems on the queenside, so with correct play it is a draw even with the pawn at h6.
Could you give an example where it would be winning due to a slightly different pawn formation?
#19
Here is an example winning due to a slightly different pawn formation
https://www.chessgames.com/perl/chessgame?gid=1494526
Do you guys know any plans associated with the likes of the rook endgame below?
It doesn't have to be this exact position, and any tips on practical rook endgames are welcome.