  Rotating star clusters near supermassive black holes are studied using
Touma-Tremaine thermodynamics of gravitationally interacting orbital ellipses.
A simple numerical procedure for calculating thermodynamic equilibrium states
for an arbitrary distribution of stars over masses and semimajor axes is
described. Spontaneous symmetry breaking and breakdown of thermodynamics at low
positive temperatures are rigorously proven for non-rotating clusters. Rotation
is introduced through a second temperature-like parameter. Both axially
symmetric and lopsided rotational equilibria are found; the lopsided equilibria
precess with the angular velocity that is given by the ratio of the two
temperatures. Eccentric stellar disc in the nucleus of Andromeda galaxy may be
an example of a lopsided thermodynamic equilibrium of a rotating black hole
star cluster. Stellar-mass black holes occupy highly eccentric orbits in
broken-symmetry star clusters, and form flattened disc-like configurations in
rotating star clusters. They are attracted to orbits that are stationary in the
frame of reference rotating with the angular velocity of the cluster. In
spherical clusters, stellar-mass black holes' orbits are significantly more
eccentric than those of the lighter stars if the temperature is negative, and
more circular if the temperature is positive. Finally we note that planets,
comets, dark matter particles and other light bodies tend to form a spherically
symmetric non-rotating sub-cluster with maximum-entropy eccentricity
distribution $\mathscr{P}(e)=2e$, even if their host cluster is rotating and
lopsided.
