  The discovery of quasars at increasingly large cosmological redshifts may
favor "direct collapse" as the most promising evolutionary route to the
formation of supermassive black holes. In this scenario, supermassive black
holes form when their progenitors - supermassive stars - become unstable to
gravitational collapse. For uniformly rotating stars supported by pure
radiation pressure and spinning at the mass-shedding limit, the critical
configuration at the onset of collapse is characterized by universal values of
the dimensionless spin and radius parameters $J/M^2$ and $R/M$, independent of
mass $M$. We consider perturbative effects of gas pressure, magnetic fields,
dark matter and dark energy on these parameters, and thereby determine the
domain of validity of this universality. We obtain leading-order corrections
for the critical parameters and establish their scaling with the relevant
physical parameters. We compare two different approaches to approximate the
effects of gas pressure, which plays the most important role, find identical
results for the above dimensionless parameters, and also find good agreement
with recent numerical results.
