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