  We investigate possibilities for a Schr\"odinger-like gravity with the
dynamical critical exponent $z=2$, where the action only contains the
first-order time derivative. The Horava gravity always admits such a relevant
deformation because the full $(d+1)$ dimensional diffeomorphism of the Einstein
gravity is replaced by the foliation preserving diffeomorphism. The dynamics is
locally trivial or topological in the pure gravity case, but we can construct a
dynamical field theory with a $z=2$ dispersion relation by introducing a
dilaton degree of freedom. Our model provides a classical starting point for
the possible quantum dilaton gravity which may be applied to a membrane
quantization.
