We review the recently found map that takes any static and neutral Kaluza-Klein black hole, i.e. any static and neutral black hole on Minkowski-space times a circle M^d × S¹, and maps it to a corresponding solution for a non- and near-extremal brane on a circle. This gives a precise connection between phases of Kaluza-Klein black holes and the thermodynamic behavior of the non-gravitational theories dual to near-extremal branes on a circle. In particular, for the thermodynamics of strongly-coupled supersymmetric Yang-Mills theories on a circle we predict the existence of a new non-uniform phase and find new information about the localized phase. We also find evidence for the existence of a new stable phase of (2, 0) Little String Theory in the canonical ensemble for temperatures above its Hagedorn temperature.