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We determine the more general geometrical flow in the space of metricscorresponding to the steepest descent for the three-dimensional gravitationalChern-Simons action, extending the results previously considered in  Class. Quantum Grav. 25 (2008) 165019, and reveling another trouble with thefour dimensional Ho\v{r}ava-Lifshitz gravity introduced in Phys. Rev. D 79,084008 (2009), and JHEP 0903, 020 (2009), which attempts to be a candidate foran UV completion of Einstein general relativity.

On the gravitational Chern-Simons action as entropy functional for three-manifolds, and the demystification of the Ho\v{r}ava-Lifshitz gravity