  The accepted idea that the expansion of the universe is accelerating needs,
for compatibility to general relativity, the introduction of some unusual forms
of matter. However, several authors have proposed that instead of making weird
hypothesis on some yet unobservable species of matter, one should follow the
original idea of the first Einstein's paper on cosmology and consider that in
the cosmic scene one has to modify the equations that controls the
gravitational metric. This possibility led us to re-examine the evolution of
the topological invariant containing two duals in a dynamical universe, the so
called Gauss-Bonnet topological invariant. The particular interest on this
invariant is due to the fact that in a homogeneous and isotropic universe this
invariant drives the cosmic acceleration. In a decelerating scenario and as a
necessary previous condition of an ulterior acceleration this invariant must
have an extremum identified to its maximum value. We will examine the
conditions for this to occur, and a description of the universe with epochs of
accelerated and decelerated expansion.
