Gauss-Bonnet Lagrangian $G\ln G$ and cosmological exact solutions

For the Lagrangian $L=G\ln G$ where $G$ is the Gauss-Bonnet curvature scalar we deduce the field equation and solve it in closed form for 3-flat Friedmann models using a state-finder parametrization. Further we show that among all Lagrangians $F(G)$ this $L$ is the only one not having the form $G^r$ with a real constant $r$ but possessing a scale-invariant field equation. This turns out to be one of its analogies to $f(R)$ theories in two-dimensional space-time. In the appendix, we systematically list several formulas for the decomposition of the Riemann tensor in arbitrary dimensions $n$, which are applied in the main deduction for $n=4$.