  We study the analytic continuation used by Lewkowycz and Maldacena to prove
the Ryu-Takayanagi formula for entanglement entropy, which is the holographic
dual of the trace of the $\beta$-power of the time evolution operator when
$\beta\in \mathbb{R}$. This will be done perturbatively by using a weakly time
dependent Hamiltonian, corresponding to a small shift of the dual static
background. Depending on the periodicity we impose on the gravitational
solution, we consider two different possibilities and compare the associated
entropies with the results obtained through a minimal area computation. To our
surprise we discover that, at first order, both choices correctly reproduce the
associated entanglement entropy. Furthermore we find unexpected divergent
contributions that we have to discard in order to fit the minimal area
computation, and an additional requirement that needs to be imposed on the
$\beta$ dependence on the metric.
