  A higher dimensional modified gravity theory with an action that includes
dimensionally continued Euler-Poincar\'e forms up to second order in curvatures
is considered. The variational field equations are derived. Matter in the
universe at large scales is modeled by a fluid satisfying an equation of state
with dimensional dichotomy. We study solutions that describe higher dimensional
steady state cosmologies with constant volume for which the three dimensional
external space is expanding at an accelerated rate while the (compact) internal
space is contracting. We showed that the second order Euler-Poincar\'e term in
the constructions of higher dimensional steady state cosmologies could be
crucial.
