A higher dimensional modified gravity theory with an action that includes dimensionally continued Euler-Poincare 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-Poincare term in the constructions of higher dimensional steady state cosmologies could be crucial.