  Shift-symmetric Horndeski theories admit an interesting class of
Schwarzschild-de Sitter black hole solutions exhibiting time-dependent scalar
hair. The properties of these solutions may be studied via a bottom-up
effective field theory (EFT) based on the background symmetries. This is in
part possible by making use of a convenient coordinate choice --
Lema\^itre-type coordinates -- in which the profile of the Horndeski scalar
field is linear in the relevant time coordinate. We construct this EFT, and use
it to understand the stability of hairy black holes in shift-symmetric
Horndeski theories, providing a set of constraints that the otherwise-free
functions appearing in the Horndeski Lagrangian must satisfy in order to admit
stable black hole solutions. The EFT is analyzed in the decoupling limit to
understand potential sources of instability. We also perform a complete
analysis of the EFT with odd-parity linear perturbations around general
spherically symmetric space-time.
