  The application of information geometric ideas to statistical mechanics using
a metric on the space of states, pioneered by Ruppeiner and Weinhold, has
proved to be a useful alternative approach to characterizing phase transitions.
Some puzzling anomalies become apparent, however, when these methods are
applied to the study of black hole thermodynamics. A possible resolution was
suggested by Quevedo et al. who emphasized the importance of Legendre
invariance in thermodynamic metrics. They found physically consistent results
for various black holes when using a Legendre invariant metric, which agreed
with a direct determination of the properties of phase transitions from the
specific heat.
  Recently, information geometric methods have been employed by Wei et al. to
study the Kehagias-Sfetsos (KS) black hole in Horava-Lifshitz gravity. The
formalism suggests that a coupling parameter in this theory plays a role
analogous to the charge in Reissner-Nordstrom (RN) black holes or angular
momentum in the Kerr black hole and calculation of the specific heat shows a
singularity which may be interpreted as a phase transition. When the curvature
of the Ruppeiner metric is calculated for such a theory it does not, however,
show a singularity at the phase transition point.
  We show that the curvature of a particular Legendre invariant ("Quevedo")
metric for the KS black hole is singular at the phase transition point. We
contrast the results for the Ruppeiner, Weinhold and Quevedo metrics and in the
latter case investigate the consistency of taking either the entropy or mass as
the thermodynamic potential.
