  Since the entropy of stationary black holes in Horndeski gravity will be
modified by the non-minimally coupling scalar field, a significant issue of
whether the Wald entropy still obeys the linearized second law of black hole
thermodynamics can be proposed. To investigate this issue, a physical process
that the black hole in Horndeski gravity is perturbed by the accreting matter
fields and finally settles down to a stationary state is considered. According
to the two assumptions that there is a regular bifurcation surface in the
background spacetime and that the matter fields always satisfy the null energy
condition, one can show that the Wald entropy monotonically increases along the
future event horizon under the linear order approximation without any specific
expression of the metric. It illustrates that the Wald entropy of black holes
in Horndeski gravitational theory still obeys the requirement of the linearized
second law. Our work strengthens the physical explanation of Wald entropy in
Horndeski gravity and takes a step towards studying the area increase theorem
in the gravitational theories with non-minimal coupled matter fields.
