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In gravitational theories where a canonical scalar field $\phi$ with apotential $V(\phi)$ is coupled to a Gauss-Bonnet (GB) term ${\cal G}$ with theLagrangian $f(\phi,{\cal G})$, we study the cosmological stability of tensorand scalar perturbations in the presence of a perfect fluid. We show that, indecelerating cosmological epochs with a positive tensor propagation speedsquared, the existence of nonlinear functions of ${\cal G}$ in $f$ alwaysinduces Laplacian instability of a dynamical scalar perturbation associatedwith the GB term. This is also the case for $f({\cal G})$ gravity, where thepresence of nonlinear GB functions $f({\cal G})$ is not allowed during theradiation- and matter-dominated epochs. A linearly coupled GB term with $\phi$of the form $\xi (\phi){\cal G}$ can be consistent with all the stabilityconditions, provided that the scalar-GB coupling is subdominant to thebackground cosmological dynamics.

Cosmological stability in $f(\phi,{\cal G})$ gravity