[en] We derive a universal thermal effective potential, which describes all possible high-temperature instabilities of the known N = 4 superstrings, using the properties of gauged N = 4 supergravity. These instabilities are due to three non-perturbative thermal dyonic modes, which become tachyonic in a region of the thermal moduli space. The latter is described by three moduli, s, t, u, which are common to all non-perturbative dual-equivalent strings with N = 4 supersymmetry in five dimensions: the heterotic on T⁴ x S¹, the type IIB on K₃ x S¹, and the type I on T⁴ x S¹. The non-perturbative instabilities are analyzed. These strings undergo a high-temperature transition to a new phase in which five-branes condense. This phase is described in detail, using both the effective supergravity and non-critical string theory in six dimensions. In the new phase, supersymmetry is perturbatively restored but broken at the non-perturbative level. In the infinite-temperature limit the theory is topological with an N = 2 supersymmetry based on a topologically non-trivial hyper-Kaehler manifold