In the United States, most professors are paid by the university on a 9 month basis for teaching (Sep - May). During the summer months a professor is typically paid from their grants and other revenue sources. How is the summer salary chosen? E.g. if a professor is paid 10kUSD/month, can (s)he choose to be paid 20kUSD/month during the summer?
I suppose this could depend on the funding agency. For the NSF, your summer salary is determined by your academic year salary and you cannot receive more than that, although you could receive less if the program officer cuts that part of your budget. I believe this is standard for U.S. funding agencies.
This situation is of course not fair to those who hold lower-paying jobs that do not reflect their achievements (but rather state budget priorities). Like many things in grant funding, there's tremendous bureaucratic inertia and no hope of changing this aspect of the system.
Under federal rules, it's not permissible to pay a faculty member on a grant at a higher rate (per hour, week, month, or whatever) than they are paid at other times of the year. It is also not permissible to use multiple sources of funding (e.g. grants from different agencies) to pay a faculty member at more than 100% of their usual pay rate.
If a faculty member has a 9 month salary of $90K, and if they have a grant with an approved budget including two months of summer salary for the faculty member, then that faculty member can be paid an additional $20K for working on the grant for two months during the summer.
Note that NSF policy does not permit faculty members to collect more than 2 months of summer salary from any combination of NSF grants. It is sometimes possible to combine funding from NSF and other sources to get a full 3 months of summer salary.