What if you only have supplies in your backpack? Will then a short rest cost you 40 pcs?

For me, it literally doesn't solve anything.
Increasing the cost of rest will in no way make the player stop using it, and more "encourage" to search every chest they come across, even if they have not done so before.
The player can buy supplies from traders and even more than double the amount of required supplies will not change anything.
Not that you need gold for anything in this game, even at the beginning of the game.
Given the age-old law of rpg, you won't even get to the middle of the game and you'll have so much gold that you won't have anything to do with it. Honestly, there hasn't been a game where at some point (usually sooner than later) you don't have millions of gold.
You might as well spend it on food.
Your solution can only really make a difference at the beginning of the game.