General vacuum solution of modified $F(G)$-gravity with Gauss-Bonnet term

In this paper we present a general vacuum solution of the modified Gauss-Bonnet gravity equations for the Friedmann-Lema\^itre-Robertson-Walker metric. We use an ansatz to reduce the gravitational equations to an ordinary differential equations for function $F=F(G)$. The solution obtained depends on an arbitrary function and is new. As an example we take an arbitrary function in the form of a power one and analyze the solutions for both $F(G)$ and the main cosmological physical quantities as scale factor, Hubble rate, the Gauss-Bonnet term and the scalar curvature.