Palatini variational principle for an extended Einstein-Hilbert action

We consider a Palatini variation on a generalized Einstein-Hilbert action. We find that the Hilbert constraint—that the connection equals the Christoffel symbol—arises only as a special case of this general action, while, for particular values of the coefficients of this generalized action, the connection is completely unconstrained. We discuss the relationship between this situation and that usually encountered in the Palatini formulation.