  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.
