The 37-Day Cooldown Period
37 days may seem like a totally random amount of time between name changes, but it is to make sure accidental name changes don't cause you to lose your precious username.
30 + 7
Everyone has to wait 30 days between name changes, but you get an extra week's worth of grace period to allow for fixing accidental name-changes or if you have second thoughts.
Is there any way to speed up this process?
Short answer: No.
It is possible to find the exact date and time when the username will become available, but there is no way to shorten the 37-day cooldown period.