This research paper examines a cosmological model in flat space-time via $f(R,G)$ gravity where $R$ and $G$ are respectively the Ricci scalar and Gauss-Bonnet invariant. Our model assumes that $f(R,G)$ is an exponential function of $G$ combined with a linear combination of $R$. We scrutinize the observational limitations under a power law cosmology that relies on two parameters - $H_0$, the Hubble constant, and $q$, the deceleration parameter, utilizing the 57-point $H(z)$ data, 8-point BAO data, 1048-point Pantheon data, joint data of $H(z)$ + Pantheon, and joint data of $H(z)$ + BAO + Pantheon. The outcomes for $H_0$ and $q$ are realistic within observational ranges. We have also addressed Energy Conditions, $Om(z)$ analysis and cosmographical parameters like Jerk, Lerk and Snap. Our estimate of $H_0$ is remarkably consistent with various recent Planck Collaboration studies that utilize the $\Lambda$CDM model. According to our study, the power law cosmology within the context of $f(R,G)$ gravity provides the most comprehensive explanation for the important aspects of cosmic evolution.

Cosmography, as an integral branch of cosmology, strives to characterize the Universe without relying on pre-determined cosmological models. This model-independent approach utilizes Taylor series expansions around the current epoch, providing a direct correlation with cosmological observations and the potential to constrain theoretical models. Various observable quantities in cosmology can be described as different combinations of cosmographic parameters. Furthermore, one can apply cosmography to models with a varying speed of light. In this case, the Hubble parameter can be expressed by the same combination of cosmographic parameters for both the standard model and varying speed of light models. However, for the luminosity distance, the two models are represented by different combinations of cosmographic parameters. Hence, luminosity distance might provide a method to constrain the parameters in varying speed-of-light models.