  By gauging the Maxwell spacetime algebra the standard geometric framework of
Einstein gravity with cosmological constant term is extended by adding six
fourvector fields A_\mu^{ab}(x) associated with the six abelian tensorial
charges in the Maxwell algebra. In the simplest Maxwell extension of Einstein
gravity this leads to a generalized cosmological term that includes a
contribution from these vector fields. We also consider going beyond the basic
gravitational model by means of bilinear actions for the new Abelian gauge
fields. Finally, an analogy with the supersymmetric generalization of gravity
is indicated. In an Appendix, we propose an equivalent description of the model
in terms of a shift of the standard spin connection by the A_\mu^{ab}(x)
fields.
