Linear perturbation analysis of hairy black holes in shift-symmetric   Horndeski theories: Odd-parity perturbations
  We analyze the mode stability of odd-parity perturbations of black holes with linearly time-dependent scalar hair in shift-symmetric Horndeski theories. We show that a large class of black hole solutions in these theories suffer from ghost or gradient instability, while there are some classes of solutions that are stable under linear odd-parity perturbations in the context of mode analysis. 
Kazufumi Takahashi, Teruaki Suyama
gr-qc hep-th