  I revisit rotating black hole solutions in three-dimensional Horava gravity
with z = 2 as a simpler set-up of the renormalizable quantum gravity `a la
Lifshitz and DeWitt. The solutions have a curvature singularity at the origin
for a non-vanishing rotation parameter J, unlike the black holes in
three-dimensional Einstein gravity. For anti-de Sitter space, there are black
hole event horizons as usual and the singularity is not naked, in agreement
with the cosmic censorship. On the other hand, for flat or de Sitter space, the
earlier solution has also a cosmic-censorship problem because there are no
conventional black hole horizons as in Einstein gravity, other than the usual
cosmological horizon for the latter case, so that the singularity could be
naked in Horava gravity. However, with the help of recent corrections, I show
that the solutions have a peculiar black hole horizon at the origin so that the
singularity is not naked even without the conventional black hole horizons in
flat or de Sitter case, due to the Lorentz-violating higher-derivative terms.
On the other hand, I note also that a new ``cosmological" horizon exists even
for the flat case, contrary to the usual wisdom, due to combined effects of the
higher derivatives and the angular-momentum barrier. I study an unified
treatment of their unusual black hole thermodynamics for the flat and de Sitter
spaces, as well as the anti-de Sitter space, which might be due to lack of the
absolute horizons in the Lorentz-violating gravity.
