Symmetries and gravitational Chern-Simons Lagrangian terms

We consider some general consequences of adding pure gravitational Chern-Simons term to manifestly diff-covariant theories of gravity, focusing essentially on spacetimes with D>3. Extending the result of a previous paper we enlarge the class of metrics for which the inclusion of a gravitational Chern-Simons term in the action does not affect solutions and corresponding physical quantities. In the case in which such solutions describe black holes (of general horizon topology) we show that the black hole entropy is also unchanged. We arrive at these conclusions by proving three general theorems and studying their consequences. One of the theorems states that the contribution of the gravitational Chern-Simons to the black hole entropy is invariant under local rescaling of the metric.