  This paper analyzes the possibility of bouncing and non-bouncing universes in
the framework of four-dimensional Einstein-Gauss-Bonnet (4D-EGB) gravity,
corresponding respectively to negative and positive coupling constants
$\lambda$ of the Gauss-Bonnet term. We also use the Horndeski-type
scalar-tensor theory to assess the role of a scalar charge $C$ as a geometrical
contribution to the radiation in the Universe. We modify the expansion history
of the universe to allow for modifications induced by the 4D-EGB gravity. Using
Planck measurements of the cosmic microwave background anisotropies as well as
various datasets of baryonic acoustic oscillations, we set the upper bounds
$\lambda \le 10^{-16} \text{(km/s/Mpc)}^{-2} $ and $\lambda \le 10^{-30}
\text{(km/s/Mpc)}^{-2} $ for the non-bouncing and bouncing scenarios. The upper
limit in the latter case is mainly driven by the requirement to conservatively
respect the thermal history at energy scales of the standard model of particle
physics. We also find that the contribution of the geometrical radiation-like
term of the model cannot exceed 10\% of the current radiation in the Universe.
The possibility of an early inflationary phase produced by a single scalar
field is also studied and found to be feasible in both bouncing and
non-bouncing scenarios. This study shows the feasibility of a bouncing
universe, even with normal matter sector, in the 4D-EGB gravity. More
theoretical investigation is required to further explore possible observational
predictions of the model that can distinguish between general relativity and
4D-EGB gravity.
