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We study static spherically symmetric black hole solutions with a linearlytime-dependent scalar field and discuss their linear stability in the shift-and reflection-symmetric subclass of quadratic degenerate higher-orderscalar-tensor (DHOST) theories. We present the explicit forms of the reducedsystem of background field equations for a generic theory within this subclass.Using the reduced equations of motion, we show that in several cases thesolution is forced to be of the Schwarzschild or Schwarzschild-(anti-)de Sitterform. We consider odd-parity perturbations around general static sphericallysymmetric black hole solutions, and derive the concise criteria for the blackholes to be stable. Our analysis also covers the case with a static or constantprofile of the scalar field.

Linear stability analysis of hairy black holes in quadratic degenerate higher-order scalar-tensor theories: Odd-parity perturbations