Topological static spherically symmetric vacuum solutions in gravity $$\mathcal{F }(R,G)$$.

The Lagrangian derivation of the Equations of Motion for topological static spherically symmetric metrics in $$\mathcal{F }(R,G)$$-modified gravity is presented and the related solutions are discussed. In particular, a new topological solution for the model $$\mathcal{F }(R,G)=R+\sqrt{G}$$ is found. The black hole solutions and the First Law of thermodynamic are analyzed. Furthermore, the coupling with electromagnetic field is also considered and a Maxwell solution is derived. [ABSTRACT FROM AUTHOR]