Great news, guys! This has been much more fully fleshed out by a staff member of the astrophysics branch of the Physics Forum.
Here is his response. It is basically what Blademan originally brought up, and this user even brought up the specific value, as being
Polytrope. Now we have a direct formula to calculate GBE of stars!
This new formula would be U = (3*G*M^2)/(r(5-n)), in which U is GBE in joules, G is the gravitational constant of 6.67408x10^-11, M is mass in kilograms, r is radius in meters, and n is the polytropic value attributed to the type of star. For example, for giants and brown dwarves would be 1.5, while n for main-sequence and degenerative core (white dwarves, for example) is 3, and extremely dense stars, such as the neutron pulsars, have an value of 1.
Using this new formula, we get a value of
5.6928564x10^41 joules for our Sun's GBE. This is closer to the original value, and actually has a new, cited, and hard-and-fast formula behind it. I believe this should be the number, and we should update our GBE page with this new polytrope-compensated formula, as well as the polytrope values for the different types of stars.
That said UY Scuti is still frail as crap.