To narrow the scope of my question, by "place" I mean rocks with a variety of routes with bolts in place, and by "cost" I refer to equipment rental and access to walls (excluding food, lodging, and transport).
I'm only an amateur climber, so perhaps I'm missing something here, but where I live (northeast USA), pretty much all climbs are openly accessible, perhaps with a small fee for parking. If you were climbing in a national park, you would have to pay an entrance fee, but nothing more than that.
As you are excluding the major costs (food, lodging, and transportation), the only cost left on your list is gear rental. And of course, the cheapest gear rental is just to buy your own gear- it's a large upfront cost, but it becomes tiny over the long term. Indeed, I would question why you would be traveling internationally to climb if you did not own your own gear already.
I think your question is too broad to be answered well but this has already been discussed on meta and apparently I was in a Christmas mood back then. However I think your question is not well-posed.
First, climbing crags (national parks and parking fees apart) are openly accessible as part of climbing philosophy. I have heard of some cases in SE Asia where owners of adjacent land tried to monetize routes that were opened (by others) and charged a fee - these places subsequently got boycotted by the local and international climbing community. With the above exceptions I would not pay for access to outdoor climbing and in fact the majority of crags I have been to has been completely free.
Second, climbing gear is what keeps you alive in the case of a fall or descend so you want it to be worthy of your trust. As has been pointed out in the other answer (to whom +1), the easiest way to ensure your survival is to buy and carry your own quality gear.
If for some reason this is not an option for you (but I do know people that backpack with climbing equipment), then I suggest you rent
- quality gear (i.e. brand names that you recognize, equipment that you personally checked to be fine as far as it is possible)
- from an established climbing store/school
Now the crucial point of your question is: how much does that cost or where is it cheapest - and my best guess is, that is going to be the same in most places. The cost of gear is the fairly the same all over the world, typically a bit more expensive in remote places but those might have cheaper infrastructure to run their store. So in the end I believe it balances out and there won't be any dramatic differences over the world.
Finally let me say that renting out climbing gear is not so common. I know it happens in some places (e.g. Tonsai, Thailand) but most stores are looking to sell and schools looking to get you on their excursions.