  In this work, we first introduce a generalized von Neumann entropy that
depends only on the density matrix. This is based on a previous proposal by one
of us modifying the Shannon entropy by considering non-equilibrium systems on
stationary states, and an entropy functional depending only on the probability.
We propose a generalization of the replica trick and find that the resulting
modified von Neumann entropy is precisely the previous mentioned entropy that
was obtained by other assumptions. Then, we address the question whether
alternative entanglement entropies can play a role in the gauge/gravity
duality. Our focus are 2d CFT and their gravity duals. Our results show
corrections to the von Neumann entropy $S_0$ that are larger than the usual
$UV$ ones and also than the corrections to the length dependent $AdS_3$ entropy
which result comparable to the $UV$ ones. The correction terms due to the new
entropy would modify the Ryu-Takayanagi identification between the CFT and the
gravitational $AdS_3$ entropies.
