in the last step we have used (\[L warped bh\]). Thus, also in this case the expected behavior for $r_{\textrm{min}} \rightarrow r_0$ is recovered (here we have $d=3$).

Perturbed Lifshitz background {#section perturbed lifshitz}
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The Lifshitz background is defined by a metric which is scale invariant if the space coordinates and the time coordinate scale with a different power. The relative scale dimension of time and space is the dynamical exponent. This parameter usually occurs in the time component of the metric; therefore it does not affect the computation of the holographic entanglement entropy, which involves the metric on a constant time slice. An example of this type is considered in the section \[section lifshitz bh\]. Instead, when the dynamical exponent occurs in some spatial component of the metric, then it usually turns out to be involved non trivially in the holographic computation of the entanglement entropy [@Azeyanagi:2009pr]. In this section we consider an example of this type.\
A perturbation of the Lifshitz background through a formal parameter expansion was studied in [@Gubser:2009cg] as a solution of the Abelian Higgs model in $AdS_4$ [@Gubser:2008px], introduced to describe superconducting black holes. The metric to consider reads $$\label{metric perturbed lifshitz}
ds^2\,=\,
