  Models of inflation in a gravitational background with an anisotropic
space-time scaling are studied. The background is a higher-dimensional Lifshitz
throat with the anisotropy scaling $z\neq 1$. After the dimensional reduction,
the four-dimensional general covariance is explicitly broken to a
three-dimensional spatial diffeomorphism. As a result the cosmological
perturbation theory in this set up with less symmetries have to be formulated.
We present the consistent cosmological perturbation theory for this set up. We
find that the effective four-dimensional gravitational wave perturbations
propagate with a different speed than the higher dimensional gravitational
excitations. Depending on the model parameters, for an observer inside the
throat, the four-dimensional gravitational wave propagation can be
superluminal. We also find that the Bardeen potential and the Newtonian
potential are different. This can have interesting observational consequences
for lensing and CMB fluctuations. Furthermore, we show that at the linearized
level the inflaton field excitations vanish.
