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Recently introduced    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\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} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\phi $$\end{document}   ϕand its kinetic term    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\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} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$f\left( \mathcal {G},\phi ,(\nabla \phi )^2,T\right) $$\end{document}    f   G  ,  ϕ  ,    (  ∇  ϕ  )   2   ,  Ttheory 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} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$f\left( \mathcal {G},\phi ,(\nabla \phi )^2,T\right) $$\end{document}    f   G  ,  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} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$f\left( \mathcal {G},\phi ,(\nabla \phi )^2,T\right) $$\end{document}    f   G  ,  ϕ  ,    (  ∇  ϕ  )   2   ,  Tgravity. 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} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$f\left( \mathcal {G},\phi ,T\right) $$\end{document}    f   G  ,  ϕ  ,  Tcase. Moreover, in order to check viability of the reconstructed models, we discuss the energy conditions in the specific cases, namely the    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$f(R,\phi ,(\nabla \phi )^2)$$\end{document}    f  (  R  ,  ϕ  ,    (  ∇  ϕ  )   2   )and    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$f=\gamma (\phi ,X)\mathcal {G}+\mu T^{1/2}$$\end{document}    f  =  γ   (  ϕ  ,  X  )   G  +  μ   T   1  /  2approaches. We show, that for the appropriate choice of parameters and constants, the energy conditions can be satisfied for the discussed scenarios.

Cosmological reconstruction and energy constraints in generalized Gauss–Bonnet-scalar–kinetic–matter couplings