  We explore the generalized covariant entropy bound in the theory where
Einstein gravity is perturbed by quadratic curvature terms, which can be viewed
as the first-order quantum correction to Einstein gravity. By replacing the
Bekenstein-Hawking entropy with the holographic entanglement entropy of this
theory and introducing two reasonable physical assumptions, we demonstrate that
the corresponding Generalized Covariant Entropy Bound is satisfied under a
first-order approximation of the perturbation from the quadratic curvature
terms. Our findings suggest that the entropy bound and the Generalized Second
Law of black holes are satisfied in the Einstein gravity under the first-order
perturbation from the quadratic curvature corrections, and they also imply that
the generalized covariant entropy bound may still hold even after considering
the quantum correction of gravity, but in this case, we may need to use
holographic entanglement entropy as the formula for gravitational entropy.
