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Abstract In general relativity (without matter), there is typically a one parameter family of static, maximally symmetric black hole solutions labeled 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, λ , of the radii of the boundary spheres. As λ approaches a critical value, λ 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.