According to the observations of Type-Ia Supernovae, the expansion of the universe is accelerating in the late-time epoch. We call the energy form which causes the accelerated expansion of the universe a dark energy. On the other hand, there is motivation from the string theory that our spacetime has had some extra directions more than three spatial dimensions in space. If spacetime has some extra dimensions in addition to the 4-dimensions that we perceive, they should be compactified and hidden from us and there must be some mechanism to stabilize them, otherwise we would have observed them in the experiments. In 2007, Greene and Levin proposed that the Casimir energy of certain combinations of massless and massive bulk fields can give a stabilizing potential for the moduli fields while driving the accelerated expansion in the 3-large spatial direction. This is the hallmark behaviour of the vacuum dark energy. In this thesis we investigate the effect of a space-like Lorentz-violating aether field with a fixed expectation value along the extra dimensional directions on the moduli stabilization mechanism. We found that the aether field can slow down the oscillations of the shape moduli fields in the universe although the non-relativistic matter is dominating.