Reconstructing cosmic expansion in $f(R,G)$ gravity using a log-periodic deceleration model

We study the late-time cosmology in $f(R,G)=R+\alpha R^2+\beta e^{\gamma G}$, using a logarithmic parametrization of the deceleration parameter $q(z)=q_0+q_{1}sin[log(1+z)]$. The Hubble parameter $H(z)$ is reconstructed and model parameters are constrained via MCMC analysis using CC ($31$), BAO ($26$) and Pantheon+SHOES ($1701$) datasets. Our results yield a Hubble constant in the range $H_0=71.7$–$72.8$ km/s/Mpc, consistent with late-time observations. The present deceleration parameter is found to be $q_0=-0.484$ to $-0.517$, while the evolution parameter $q_1\approx 1$, indicating increasing acceleration. The transition redshift shifts from $z_{tr}=0.879$ (CC) to $0.744$ (CC+BAO+Pantheon+SHOES), supporting a dynamic acceleration phase. The model reproduces early radiation behavior with $\omega (z>>1)\approx 0.33$ and predicts present-day values ${\omega }_0\approx -0.49$. Energy conditions NEC and DEC are satisfied, while SEC is violated at late times. The statefinder parameters $\{r_0,s_0\}=(0.866,0.046)$ lie near the $\mathrm{\Lambda }$CDM point. Estimated age of the Universe ranges from $13.01$ to $13.59$ Gyr. Thermodynamic analysis confirms consistency with the generalized second law. Overall, the model offers a viable and observationally consistent description of cosmic acceleration.