Entropy vs Gravitational Action: Do Total Derivatives Matter?

The total derivatives in the gravitational action are usually disregarded as non-producing any non-trivial dynamics. In the context of the gravitational entropy, within Wald's approach, these terms are considered irrelevant as non-contributing to the entropy. On the other hand, the total derivatives are usually present in the trace anomaly in dimensions higher than 2. As the trace anomaly is related to the logarithmic term in the entanglement entropy it is natural to ask whether the total derivatives make any essential contribution to the entropy or they can be totally ignored. In this note we analyze this question for some particular examples of total derivatives. Rather surprisingly, in all cases that we consider the total derivatives produce non-trivial contributions to the entropy. Some of them are non-vanishing even if the extrinsic curvature of the surface is zero. We suggest that this may explain the earlier observed discrepancy between the holographic entanglement entropy and Wald's entropy.