  The non-perturbative instabilities of hot Kaluza-Klein spacetime are
investigated. In addition to the known instability of hot space (the nucleation
of 4D black holes) and the known instability of KK space (the nucleation of
bubbles of nothing by quantum tunneling), we find two new instabilities: the
nucleation of 5D black holes, and the nucleation of bubbles of nothing by
thermal fluctuation. These four instabilities are controlled by two Euclidean
instantons, with each instanton doing double duty via two inequivalent analytic
continuations; thermodynamic instabilities of one are shown to be related to
mechanical instabilities of the other. I also construct bubbles of nothing that
are formed by a hybrid process involving both thermal fluctuation and quantum
tunneling. There is an exact high-temperature/low-temperature duality that
relates the nucleation of black holes to the nucleation of bubbles of nothing.
