Abstract
Recently introduced \documentclass[12pt]{minimal}
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\begin{document}$$f(\mathcal {G},T)$$\end{document}f(G,T) theory is generalized by adding dependence on the arbitrary scalar field \documentclass[12pt]{minimal}
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\begin{document}$$\phi $$\end{document}ϕ and its kinetic term \documentclass[12pt]{minimal}
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\begin{document}$$(\nabla \phi )^2$$\end{document}(∇ϕ)2, to explore non-minimal interactions between geometry, scalar and matter fields in context of the Gauss–Bonnet theories. The field equations for the resulting \documentclass[12pt]{minimal}
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\begin{document}$$f\left( \mathcal {G},\phi ,(\nabla \phi )^2,T\right) $$\end{document}fG,ϕ,(∇ϕ)2,T theory are obtained and show that particles follow non-geodesic trajectories in a perfect fluid surrounding. The energy conditions in the Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime are discussed for the generic function \documentclass[12pt]{minimal}
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\begin{document}$$f\left( \mathcal {G},\phi ,(\nabla \phi )^2,T\right) $$\end{document}fG,ϕ,(∇ϕ)2,T. As an application of the introduced extensions, using the reconstruction techniques we obtain functions that satisfy common cosmological models, along with the equations describing energy conditions for the reconstructed \documentclass[12pt]{minimal}
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\begin{document}$$f\left( \mathcal {G},\phi ,(\nabla \phi )^2,T\right) $$\end{document}fG,ϕ,(∇ϕ)2,T gravity. The detailed discussion of the energy conditions for the de Sitter and power-law spacetimes is provided in terms of the fixed kinetic term i.e. in the \documentclass[12pt]{minimal}
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\begin{document}$$f\left( \mathcal {G},\phi ,T\right) $$\end{document}fG,ϕ,T case. Moreover, in order to check viability of the reconstructed models, we discuss the energy conditions in the specific cases, namely the \documentclass[12pt]{minimal}
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\begin{document}$$f(R,\phi ,(\nabla \phi )^2)$$\end{document}f(R,ϕ,(∇ϕ)2) and \documentclass[12pt]{minimal}
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\begin{document}$$f=\gamma (\phi ,X)\mathcal {G}+\mu T^{1/2}$$\end{document}f=γ(ϕ,X)G+μT1/2 approaches. We show, that for the appropriate choice of parameters and constants, the energy conditions can be satisfied for the discussed scenarios.
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