  We argue that the configurations that approach maximal entropy in
five-dimensional asymptotically flat vacuum gravity, for fixed mass and angular
momentum, are `black Saturns' with a central, close to static, black hole and a
very thin black ring around it. For any value of the angular momentum, the
upper bound on the entropy is equal to the entropy of a static black hole of
the same total mass. For fixed mass, spin and area there are families of
multi-ring solutions with an arbitrarily large number of continuous parameters,
so the total phase space is infinite-dimensional. Somewhat surprisingly, the
phases of highest entropy are not in thermal equilibrium. Imposing
thermodynamical equilibrium drastically reduces the phase space to a finite,
small number of different phases.
