De Sitter solutions play an important role in cosmology because the knowledge of unstable de Sitter solutions can be useful to describe inflation, whereas stable de Sitter solutions are often used in models of late-time acceleration of the Universe. Some models with scalar fields have both stable and unstable de Sitter solutions that correspond to different fixed values of the scalar field. The models the Gauss-Bonnet term are actively used both as inflationary models and as dark energy models. To modify the Einstein equations one can add a nonlinear function of the Gauss-Bonnet term or a function of the scalar field multiplied on the Gauss-Bonnet term.

In order to find out the de Sitter solutions in a model with a minimally coupled scalar field with a potential $V$ it is enough to find zeros of the first derivative of $V$. The sign of the second derivative of $V$ at a de Sitter point determines the stability of the solution. The effective potential plays the same role in the gravity models with the Gauss-Bonnet term because the stable de Sitter solutions correspond to minima of the effective potential.

So, the effective potential method essentially simplifies the search and stability analysis of de Sitter solutions. My talk is based on the paper by E.O. Pozdeeva, M. Sami, A.V. Toporensky, and S.Yu. Vernov, Phys. Rev. D 100 (2019) 083527, and recent investigations.