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In the context of the holographic duality, the entanglement entropy ofordinary QFT in a subregion in the boundary is given by a quarter of the areaof an minimal surface embedded in the bulk spacetime. This rule has been alsoextended to a suitable one-parameter generalization of the von-Neuman entropy$\hat{S}_n$ that is related to the R\'enyi entropies $S_n$, as given by thearea of a \emph{cosmic brane} minimally coupled with gravity, with a tensionrelated to $n$ that vanishes as $n\to1$, and moreover, this parameter can beanalytically extended to arbitrary real values. However, the brane action playsno role in the duality and cannot be considered a part of the theory ofgravity, thus it is used as an auxiliary tool to find the correct backgroundgeometry.  In this work we study the construction of the gravitational (reduced) densitymatrix from holographic states, whose wave-functionals are described aseuclidean path integrals with arbitrary conditions on the asymptoticboundaries, and argue that in general, a non-trivial Hayward term must be haveninto account. So we propose that the gravity model with a coupled Nambu-Gotoaction is not an artificial tool to account for the R\'enyi entropies, but itis present in the own gravity action through a Hayward term. As a result weshow that the computations using replicas simplify considerably and we recoverthe holographic prescriptions for the measures of entanglement entropy; inparticular, derive an area law for the original R\'enyi entropies ($S_n$)related to a minimal surface in the $n$ replicated spacetime. Moreover, we showthat the gravitational modular flow contains the area operator and can explainthe Jafferis-Lewkowycz-Maldacena-Suh proposal.

R\'enyi entropies and area operator from gravity with Hayward term