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In the present manuscript the basic Einstein--Hilbert cosmological model isextended, by adding a new functional $F(G, T_{\mu\nu}T^{\mu\nu})$ in thefundamental action, encoding specific geometrical effects due to a nontrivialcoupling with the Gauss-Bonnet invariant ($G$), and the energy--momentumsquared term ($T_{\mu\nu}T^{\mu\nu}$). After obtaining the correspondinggravitational field equations for the specific decomposition where $F(G,T_{\mu\nu}T^{\mu\nu})=f(G)+g(T_{\mu\nu}T^{\mu\nu})$, we have explored thephysical features of the cosmological model by considering the linear stabilitytheory, an important analytical tool in the cosmological theory which canreveal the dynamical characteristics of the phase space. The analyticalexploration of the corresponding phase space structure revealed that thepresent model can represent a viable dark energy model, with various stationarypoints where the effective equation of state corresponds to a de--Sitter epoch,possible explaining the early and late time acceleration of the Universe.

The accelerated expansion in $F(G,T_{\mu \nu}T^{\mu \nu})$ gravity