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The first law of black hole thermodynamics is suitable for any diffeomorphisminvariant gravity, and the entropy in the first law is the Wald entropy whichis highly dependent on the non-minimal coupling interactions in the theory ofgravity. However, whether the Wald entropy still satisfies the second law needsto be investigated. The entropy of black holes obeying the linearized secondlaw in arbitrary high-order curvature gravity is given, which can be written asthe Wald entropy with correction terms. It indicates that the Wald entropy isnot commonly obeying the linearized second law for any high-order curvaturegravity. When the interactions of gravity with matter fields are included inthe theory of gravity, the entropy of black holes obeying the linearized secondlaw has not been obtained in this case. Considering any high-order curvaturegravity with the scalar and the electromagnetic fields, from the Raychaudhuriequation, the entropy obeying the linearized second law is generally obtained,which can be expressed as the Wald entropy with correction terms as well. Theentropy does not include the contribution from the electromagnetic fields, andthe correction terms contain the contribution from the minimal couplinginteraction between gravity and the scalar fields. Since the entropy satisfyingthe linearized second law depends only on the non-minimal coupling interactionof gravity in previous research, this result upends our understanding of theentropy of black holes obeying the linearized second law in any gravitationaltheory with matter fields.

The linearized second law for any higher curvature gravity with the scalar and the electromagnetic fields