Finite-time singularities in modified $\mathcal{F}(R,G)$-gravity and singularity avoidance

We study finite-time future singularities in $\mathcal{F}(R,G)$-gravity, where $R$ and $G$ are the Ricci scalar and the Gauss-Bonnet invariant, respectively. In particular, we reconstruct the $F(G)$-gravity and $\mathcal{F}(R,G)$-gravity models realizing the finite-time future singularities. We discuss a possible way to cure the finite-time future singularities in $\mathcal{F}(R,G)$-gravity by taking into account higher-order curvature corrections or effects of viscous fluids.