This work is concerned to study the bouncing nature of the universe for an isotropic configuration of fluid T αβ and Friedmann-Lemaître-Robertson-Walker metric scheme. This work is carried out under the novel f (G, T αβ T αβ ) gravitation by assuming a specific model i.e, f (G, T 2 ) = G + αG 2 + 2λT 2 with α and λ are constants, serving as free parameters. The terms G and T 2 served as an Gauss-Bonnet invariant and square of the energy-momentum trace term as an inclusion in the gravitational action respectively, and is proportional to T 2 = T αβ T αβ . A specific functional form of the Hubble parameter is taken to provide the evolution of cosmographic parameters. A well known equation of state parameter, ω(t) = − k log(t+ǫ) t − 1 is used to represent the dynamical behavior of energy density, matter pressure and energy conditions. A detailed graphical analysis is also provided to review the bounce. Furthermore, all free parameters are set in a way, to make the supposed Hubble parameter act as the bouncing solution and ensure the viability of energy conditions. Conclusively, all necessary conditions for a bouncing model are checked.