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

Accelerated Expansion of the Universe in a Higher Dimensional Modified Gravity with Euler-Poincar\'e Terms