Bouncing Cosmology in f ( R , G ) $f(R,\mathcal {G})$ Gravity with Thermodynamic Analysis

The evolution of the universe in a modified gravity theory that includes the terms Ricci scalar ($R$) and the Gauss-Bonnet invariant ($G$) is studied. The function $f(R,G)$ is obtained using the e-folding number and reconstruction technique by assuming an appropriate parameterization of the scale factor $a$. In this model, the various cosmological parameters are analyzed to explicate the bouncing scenario of the universe with the help of the contraction and expansion phases of the universe before and after the bouncing point of the model, respectively. A violation of the null energy condition is found. Additionally, the ghost condensate nature of the model in the neighborhood of the bouncing point ($t=0$) is seen. Furthermore, the deceleration parameter is not defined at the bouncing point, i.e., $q\to \infty$ at $t=0$ and approaches a finite value $q\to -1+\frac{1}{\beta }$ in later times ($t\to \infty$). Finally, the evolution of the Hawking temperature and the validity of the second law of thermodynamics in the bouncing scenario of our model are discussed.