The paper proposes a new formula for the entropy of a dynamical black hole in an arbitrary classical diffeomorphism covariant Lagrangian theory of gravity in \( n \) dimensions. This formula agrees with the usual Noether charge formula for stationary black holes but includes a nontrivial correction term for nonstationary black holes. In general relativity, the formula differs from the standard Bekenstein–Hawking entropy by a term involving the integral of the expansion of the null generators of the horizon. The authors show that this dynamical entropy in general relativity is equal to \( \frac{A[\mathcal{C}]}{4} - \frac{1}{4} \int_{\mathcal{C}} V \partial \), where \( A[\mathcal{C}] \) is the area of the cross-section \( \mathcal{C} \), \( V \) is an affine parameter of the null generators, and \( \partial \) is the expansion of these generators. The formula is derived from the requirement that a "local physical process version" of the first law of black hole thermodynamics holds for perturbations of a stationary black hole. The entropy obeys the second law of black hole thermodynamics for first-order perturbations sourced by external matter satisfying the null energy condition. For vacuum perturbations, the leading-order change in entropy occurs at second order, and the second law is obeyed if and only if the modified canonical energy flux is positive. The paper also discusses the relationship between the proposed formula and the Dong–Wall entropy formula, showing that they are closely related. Additionally, it analyzes the generalized second law in semiclassical gravity, demonstrating that the quantum null energy condition (QNEC) on a Killing horizon is equivalent to the generalized second law using the authors' notion of black hole entropy, while the generalized second law for the Dong–Wall entropy is equivalent to an integrated version of QNEC.The paper proposes a new formula for the entropy of a dynamical black hole in an arbitrary classical diffeomorphism covariant Lagrangian theory of gravity in \( n \) dimensions. This formula agrees with the usual Noether charge formula for stationary black holes but includes a nontrivial correction term for nonstationary black holes. In general relativity, the formula differs from the standard Bekenstein–Hawking entropy by a term involving the integral of the expansion of the null generators of the horizon. The authors show that this dynamical entropy in general relativity is equal to \( \frac{A[\mathcal{C}]}{4} - \frac{1}{4} \int_{\mathcal{C}} V \partial \), where \( A[\mathcal{C}] \) is the area of the cross-section \( \mathcal{C} \), \( V \) is an affine parameter of the null generators, and \( \partial \) is the expansion of these generators. The formula is derived from the requirement that a "local physical process version" of the first law of black hole thermodynamics holds for perturbations of a stationary black hole. The entropy obeys the second law of black hole thermodynamics for first-order perturbations sourced by external matter satisfying the null energy condition. For vacuum perturbations, the leading-order change in entropy occurs at second order, and the second law is obeyed if and only if the modified canonical energy flux is positive. The paper also discusses the relationship between the proposed formula and the Dong–Wall entropy formula, showing that they are closely related. Additionally, it analyzes the generalized second law in semiclassical gravity, demonstrating that the quantum null energy condition (QNEC) on a Killing horizon is equivalent to the generalized second law using the authors' notion of black hole entropy, while the generalized second law for the Dong–Wall entropy is equivalent to an integrated version of QNEC.