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

Recently introduced [Formula: see text] theory is generalized by adding dependence on the arbitrary scalar field [Formula: see text] and its kinetic term [Formula: see text], to explore non-minimal interactions between geometry, scalar and matter fields in context of the Gauss-Bonnet theories. The field equations for the resulting [Formula: see text] 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 [Formula: see text]. 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 [Formula: see text] 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 [Formula: see text] case. Moreover, in order to check viability of the reconstructed models, we discuss the energy conditions in the specific cases, namely the [Formula: see text] and [Formula: see text] approaches. We show, that for the appropriate choice of parameters and constants, the energy conditions can be satisfied for the discussed scenarios.