I remember reading, a long time ago, about some kind of fee (with a classically misleading name) that other countries must pay per mile to the US when flying over the Pacific. Or maybe it was sailing? Anyway, Google turns up nothing. Maybe I imagined it.
The FAA charges airlines a fee whenever they fly through U.S. administered airspace, it's about $27 per hundred nautical miles flown over the ocean and double that if they go over land.
In classic American fashion, however, the portion of the Pacific Ocean whose airspace the U.S. administers is ridiculous. Meaning a lot of airlines have to pay the FAA even if their flights go nowhere near actual U.S. territory.
I mean, you don't want it to be a free-for-all or anything and the US does have the most developed aviation infrastructure, so they're qualified to "administer" it. When used for commercial purposes it probably makes sense for various companies to contribute to those costs.
Of course, the US would never forego a power projection opportunity in the spirit of international cooperation, so the sensible solution of admisinstering ATC over the Pacific using an international organization will never happen.