  We calculate the thermal partition function in the canonical ensemble for
type IIB superstrings in the plane wave background with constant null R-R
5-form. The Hagedorn temperature is found to be higher than the corresponding
value for strings in flat space. In the limit corresponding to the weakly
coupled field theory we find that the Hagedorn temperature is pushed to
infinity. The key property of strings in the plane wave background under
investigation on which our result relies is that the effective mass of the
bosonic and fermionic coordinates in the light-cone gauge is proportional to
the momentum $p^+$. The free energy is finite as the Hagedorn temperature is
approached from below, suggesting a possible phase transition.
