  We investigate a possibility for construction of the conventional Friedmann
cosmology for our observable Universe if underlying theory is multidimensional
Kaluza-Klein model endowed with a perfect fluid. We show that effective
Friedmann model obtained by dynamical compactification of the multidimensional
one is faced with too strong variations of the fundamental "constants". From
other hand, models with stable compactification of the internal space are free
from this problem and also result in conventional 4D cosmological behavior for
our Universe. We prove a no-go theorem which shows that stable compactification
of the internal spaces is possible only if equations of state in the external
and internal spaces are properly adjusted to each other. With a proper choice
of parameters (fine tuning), effective cosmological constant in this model
provides the late time acceleration of the Universe. The fine tuning problem is
resolved in the case of the internal spaces in the form of orbifolds with
branes in fixed points. However, in this case the effective potential is too
flat (mass gravexcitons is very small) to provide necessary constancy of the
effective fundamental "constants".
