  We construct higher-derivative gravity theories in three dimensions that
admit holographic $c$-theorems and exhibit a unique maximally symmetric vacuum,
at arbitrary order $n$ in the curvature. We show that these theories exhibit
special properties, the most salient ones being the decoupling of ghost modes
around Anti-de Sitter (AdS) space, the enhancement of symmetries at linearized
level, and the existence of a one-parameter generalization of the
Ba\~nados-Teitelboim-Zanelli (BTZ) black hole that, while being asymptotically
AdS, is not of constant curvature but rather exhibits a curvature singularity.
For such black holes, we provide a holographic derivation of their
thermodynamics. This gives a microscopic picture of black hole thermodynamics
for non-supersymmetric solutions, of non-constant curvature in
higher-derivative theories of arbitrary order in the curvature.
