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We study odd parity perturbations of spherically symmetric black holes withtime-dependent scalar hair in shift-symmetric higher-order scalar-tensortheories. The analysis is performed in a general way without assuming thedegeneracy conditions. Nevertheless, we end up with second-order equations fora single master variable, similarly to cosmological tensor modes. We thusidentify the general form of the quadratic Lagrangian for the odd parityperturbations, leading to a generalization of the Regge-Wheeler equation. Wealso investigate the structure of the effective metric for the master variableand refine the stability conditions. As an application of our generalizedRegge-Wheeler equation, we compute the quasi-normal modes of a certainnontrivial black hole solution. Finally, our result is extended to include thematter energy-momentum tensor as a source term.

Perturbations and quasi-normal modes of black holes with time-dependent scalar hair in shift-symmetric scalar-tensor theories