  The first law of black hole thermodynamics is suitable for any diffeomorphism
invariant gravity, and the entropy in the first law is the Wald entropy which
is highly dependent on the non-minimal coupling interactions in the theory of
gravity. However, whether the Wald entropy still satisfies the second law needs
to be investigated. The entropy of black holes obeying the linearized second
law in arbitrary high-order curvature gravity is given, which can be written as
the Wald entropy with correction terms. It indicates that the Wald entropy is
not commonly obeying the linearized second law for any high-order curvature
gravity. When the interactions of gravity with matter fields are included in
the theory of gravity, the entropy of black holes obeying the linearized second
law has not been obtained in this case. Considering any high-order curvature
gravity with the scalar and the electromagnetic fields, from the Raychaudhuri
equation, the entropy obeying the linearized second law is generally obtained,
which can be expressed as the Wald entropy with correction terms as well. The
entropy does not include the contribution from the electromagnetic fields, and
the correction terms contain the contribution from the minimal coupling
interaction between gravity and the scalar fields. Since the entropy satisfying
the linearized second law depends only on the non-minimal coupling interaction
of gravity in previous research, this result upends our understanding of the
entropy of black holes obeying the linearized second law in any gravitational
theory with matter fields.
