Thermodynamical functions are determined for the bosonic gas distribution in the external gravitational field described by multidimensional cosmological models having the structure FRW B, where B is any D-dimensional compact space with the scalar curvature R. Universal asymptotics of this function are found for the following situations: 1) at high temperatures, with R 1 and 1, 2) at low temperatures, with R 1 and 1, where R is the scale factor of the physical space, and is the reciprocal of temperature. It is shown that if R / R 1 (where R and R are curvature scalars of macro- and microspace, correspondingly), the Casimir energy is always negative. These results are applied to discuss the dimensional reduction generated by the classical Einstein equations with quantum corrections. The idea of the dynamical dimensional reduction is expressed, in terms of the dynamical system theory, as the problem of the existence of a single stable critical point representing a configuration with the static internal space. It is demonstrated that, in the low-temperature approximation, there is no effective mechanism of the dimensional reduction to a static internal space, if B is a group manifold (with the same scale factor in all internal directions). On the other hand, the effective mechanism of the dimensional reduction to the zero size does exist. The existence of such mechanism for the full class of multidimensional homogeneous cosmologies with the hydrodynamic energy-momentum tensor is also discussed.