  We derive the equations of motion of an action-dependent version of the
Einstein-Hilbert Lagrangian, as a specific instance of the Herglotz variational
problem. Action-dependent Lagrangians lead to dissipative dynamics, which
cannot be obtained with the standard method of Lagrangian field theory.
First-order theories of this kind are relatively well understood, but examples
of singular or higher-order action-dependent field theories are scarce. This
work constitutes an example of such a theory. By casting the problem in clear
geometric terms we are able to obtain a Lorentz invariant set of equations,
which contrasts with previous attempts.
