  In general relativity (without matter), there is typically a one parameter
family of static, maximally symmetric black hole solutions labelled by their
mass. We show that there are situations with many more black holes. We study
asymptotically anti-de Sitter solutions in six and seven dimensions having a
conformal boundary which is a product of spheres cross time. We show that the
number of families of static, maximally symmetric black holes depends on the
ratio, $\lambda$, of the radii of the boundary spheres. As $\lambda$ approaches
a critical value, $\lambda_{c}$, the number of such families becomes infinite.
In each family, we can take the size of the black hole to zero, obtaining an
infinite number of static, maximally symmetric non-black hole solutions. We
discuss several applications of these results, including Hawking-Page phase
transitions and the phase diagram of dual field theories on a product of
spheres, new positive energy conjectures, and more.
