This paper proposes a new formula for the entropy of a dynamical black hole, valid to leading order for perturbations off a stationary black hole background in an arbitrary classical diffeomorphism covariant gravity theory. In stationary eras, this formula agrees with the Noether charge formula, but in nonstationary eras, it includes a nontrivial correction term. In general relativity, the formula differs from the Bekenstein-Hawking formula by a term involving the integral of the expansion of the null generators of the horizon. 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. It is shown that for first order perturbations sourced by external matter satisfying the null energy condition, the entropy obeys the second law of black hole thermodynamics. For vacuum perturbations, the leading order change in entropy occurs at second order, and the second law is obeyed at leading order if the modified canonical energy flux is positive. The formula differs from those proposed independently by Dong and Wall. The paper also considers the generalized second law in semiclassical gravity for first order perturbations of a stationary black hole, showing that the validity of the quantum null energy condition (QNEC) on a Killing horizon is equivalent to the generalized second law using the proposed entropy and a modified von Neumann entropy for matter. The paper analyzes the relationship between the proposed entropy and the Dong-Wall entropy, showing a close relationship. The paper also discusses the generalized second law and its relationship to QNEC for both entropies. The paper concludes with a discussion of the implications of the results for the understanding of black hole thermodynamics in general relativity and other gravity theories.This paper proposes a new formula for the entropy of a dynamical black hole, valid to leading order for perturbations off a stationary black hole background in an arbitrary classical diffeomorphism covariant gravity theory. In stationary eras, this formula agrees with the Noether charge formula, but in nonstationary eras, it includes a nontrivial correction term. In general relativity, the formula differs from the Bekenstein-Hawking formula by a term involving the integral of the expansion of the null generators of the horizon. 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. It is shown that for first order perturbations sourced by external matter satisfying the null energy condition, the entropy obeys the second law of black hole thermodynamics. For vacuum perturbations, the leading order change in entropy occurs at second order, and the second law is obeyed at leading order if the modified canonical energy flux is positive. The formula differs from those proposed independently by Dong and Wall. The paper also considers the generalized second law in semiclassical gravity for first order perturbations of a stationary black hole, showing that the validity of the quantum null energy condition (QNEC) on a Killing horizon is equivalent to the generalized second law using the proposed entropy and a modified von Neumann entropy for matter. The paper analyzes the relationship between the proposed entropy and the Dong-Wall entropy, showing a close relationship. The paper also discusses the generalized second law and its relationship to QNEC for both entropies. The paper concludes with a discussion of the implications of the results for the understanding of black hole thermodynamics in general relativity and other gravity theories.