Using the properties of gauged N=4 supergravity, we show that it is possible
to derive a universal thermal effective potential that describes all possible
high-temperature instabilities of the known N=4 superstrings. These
instabilities are due to non-perturbative dyonic modes, which become tachyonic
in a region of the thermal moduli space M={s,t,u}; M is common to all
non-perturbative dual-equivalent N=4 superstrings in five dimensions. We
analyse the non-perturbative thermal potential and show the existence of a
phase transition at high temperatures corresponding to a condensation of
5-branes. This phase is described in detail, using an effective non-critical
string theory.Comment: 15 pages, LATEX file, no figure
