  For binary systems of corotating black holes, the zeroth law of black hole
mechanics states that the surface gravity is constant over each component of
the horizon. Using the approximation of a conformally flat spatial metric, we
compute sequences of quasi-equilibrium initial data for corotating black hole
binaries with irreducible mass ratios in the range 10:1-1:1. For orbits outside
the innermost stable one, the surface gravity is found to be constant on each
component of the apparent horizon at the sub-percent level. We compare those
numerical results to the analytical predictions from post-Newtonian theory at
the fourth (4PN) order and from black hole perturbation theory to linear order
in the mass ratio. We find a remarkably good agreement for all mass ratios
considered, even in the strong-field regime. In particular, our findings
confirm that the domain of validity of black hole perturbative calculations
appears to extend well beyond the extreme mass-ratio limit.
