  Considering the so-called Ricci-based gravity theories, a family of
extensions of General Relativity whose action is given by a non-linear function
of contractions and products of the (symmetric part of the) Ricci tensor of an
independent connection, the Hamiltonian formulation of the theory is obtained.
To do so, the independent connection is decomposed in two parts, one compatible
with a metric tensor and the other one given by a 3-rank tensor. Subsequently,
the Riemann tensor is expressed in terms of its projected components onto a
hypersurface, allowing to construct the $3+1$ decomposition of the theory and
the corresponding Gauss-Codazzi relations, where the boundary terms naturally
arise in the gravitational action. Finally, the ADM decomposition is followed
in order to construct the corresponding Hamiltonian and the ADM energy for any
Ricci-based gravity theory. The formalism is applied to the simple case of
Schwarzschild space-time.
