  We consider a modification of the standard Einstein theory in four
dimensions, alternative to R. Jackiw and S.-Y. Pi, Phys. Rev. D 68, 104012
(2003), since it is based on the first-order (Einstein-Cartan) approach to
General Relativity, whose gauge structure is manifest. This is done by
introducing an additional topological term in the action which becomes a
Lorentz-violating term by virtue of the dependence of the coupling on the
space-time point. We obtain a condition on the solutions of the Einstein
equations, such that they persist in the deformed theory, and show that the
solutions remarkably correspond to the classical solutions of a collection of
independent 2+1-d (topological) Chern-Simons gravities. Finally, we study the
relation with the standard second-order approach and argue that they both
coincide to leading order in the modulus of the Lorentz-violating vector field.
