  We investigate the black hole (BH) scaling relation in galaxies using a model
in which the galaxy halo and central BH are a self-gravitating sphere of dark
matter (DM) with an isotropic, adiabatic equation of state. The equipotential
where the escape velocity approaches the speed of light defines the horizon of
the BH. We find that the BH mass ($m_\bullet$) depends on the DM entropy, when
the effective thermal degrees of freedom ($F$) are specified. Relations between
BH and galaxy properties arise naturally, with the BH mass and DM velocity
dispersion following $m_\bullet\propto\sigma^{F/2}$ (for global mean density
set by external cosmogony). Imposing observationally derived constraints on $F$
provides insight into the microphysics of DM. Given that DM velocities and
stellar velocities are comparable, the empirical correlation between
$m_\bullet$ and stellar velocity dispersions $\sigma_\star$ implies that
$7<F<10$. A link between $m_\bullet$ and globular cluster properties also
arises because the halo potential binds the globular cluster swarm at large
radii. Interestingly, for $F>6$ the dense dark envelope surrounding the BH
approaches the mean density of the BH itself, while the outer halo can show a
nearly uniform kpc-scale core resembling those observed in galaxies.
