I know that in 3D, there is a gravitational CS action given by

$$S = \int d^3 x \, \sqrt{-g} \, \epsilon^{\mu\nu\rho} \Big( R\_{\mu\nu}{}^{ab} \omega\_{\rho ab} + \frac23 \omega\_{\mu}{}^{ab} \omega\_{\nu b c}\omega\_\rho{}^c{}\_a \Big) \,, $$

whose field equation read the Cotton tensor

$0 = C\_{\mu\nu} = \epsilon\_\mu{}^{\tau\rho} \nabla\_\tau S\_{\rho \nu}$ ,

where $S\_{\mu\nu}$ is the Schouten tensor

$S\_{\mu\nu} = R\_{\mu\nu} - \frac14 g\_{\mu\nu} R \,.$

I am wondering if there is a five dimensional gravitational Chern-Simons action, and does it have a field equation that is defined by a special tensor such as the Cotton tensor?