Higher-order Gauss-Bonnet cosmology by Lagrange multipliers

We study cosmological models derived from higher-order Gauss-Bonnet gravity F( R, G) by using the Lagrange multiplier approach without assuming the presence of additional fields with the exception of standard perfect fluid matter. The presence of Lagrange multipliers reduces the number of allowed solutions. We need to introduce compatibility conditions of the FRW equations, which impose strict restrictions on the metric or require the introduction of additional exotic matter. Several classes of F( R, G) models are generated and discussed.