For a Horndeski-like scalar-tensor model with a chosen background solution, scalar stability and entropy-production bounds restrict the coupling gamma to be non-positive and constrain gamma*Lambda relative to alpha.
(31) As commented on in [7] the sum of the entropy Sin and that of the apparent horizon Sh ≡ S, denoted Ssum = Sin + Sh, should be considered as well
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Cosmological scalar perturbations in Horndeski-like gravity
For a Horndeski-like scalar-tensor model with a chosen background solution, scalar stability and entropy-production bounds restrict the coupling gamma to be non-positive and constrain gamma*Lambda relative to alpha.