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≠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 setup with less symmetries have to be formulated. We present the consistent cosmological perturbation theory for this setup. 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 cosmic microwave background fluctuations. Furthermore, we show that at the linearized level the inflaton field excitations vanish.