  Following on from recent studies of string theory on a one-parameter family
of integrable deformations of $AdS_{5}\times S^{5}$ proposed by Lunin and
Maldacena, we carry out a systematic analysis of the high temperature
properties of type IIB strings on the associated pp-wave geometries. In
particular, through the computation of the thermal partition function and free
energy we find that not only does the theory exhibit a Hagedorn transition in
both the $(J,0,0)$ and $(J,J,J)$ class of pp-waves, but that the Hagedorn
temperature is insensitive to the deformation suggesting an interesting
universality in the high temperature behaviour of the pp-wave string theory. We
comment also on the implications of this universality on the
confinement/deconfinement transition in the dual $\mathcal{N}=1$
Leigh-Strassler deformation of ${\cal N}=4$ Yang-Mills theory.
