  We investigate various cosmological aspects of a 4-Dimensional Gauss-Bonnet
Lagrangian, which is integrated into the Einstein Lagrangian with an arbitrary
sign, using the Friedman-Lema\^itre-Robertson-Walker (FLRW) metric. We consider
a general potential term, $V(a)$, that depends on the scale factor $a$, and we
analyze several scenarios by investigating the critical points of the dynamical
equations and stability conditions to understand how the universe's behavior is
affected by the Gauss-Bonnet term. Our research suggests that choosing the
negative sign, this integration allows for the spontaneous breaking of time
reflection symmetry. This can lead to the generation of a bounce universe even
with a normal matter sector, marking a significant departure from traditional
theories. Furthermore, we examine the possibility of a time-crystal universe,
showing that under certain circumstances, the theory might give rise to cyclic
universes.
