  We consider a higher derivative gravity theory in four dimensions with a
negative cosmological constant and show that vacuum solutions of both Lifshitz
type and Schr\"{o}dinger type with arbitrary dynamical exponent z exist in this
system. Then we find an analytic black hole solution which asymptotes to the
vacuum Lifshitz solution with z=3/2 at a specific value of the coupling
constant. We analyze the thermodynamic behavior of this black hole and find
that the black hole has zero entropy while non-zero temperature, which is very
similar to the case of BTZ black holes in new massive gravity at a specific
coupling. In addition, we find that the three dimensional Lifshitz black hole
recently found by E. Ayon-Beato et al. has a negative entropy and mass when the
Newton constant is taken to be positive.
