the holographic entanglement entropy then gives $$S_A\,=\,
\frac{\textrm{Area}(\gamma_A)}{4 G_N^{(D+1)}}\,=\,
\frac{2L_\perp^{d-1}}{4 G_N^{(d+2)}}
\int_{r_{\textrm{min}}}^{\alpha} 
\frac{B(r)^{2d-1}}{A(r)\sqrt{B(r)^{2d}-B(r_{\textrm{min}})^{2d}}}\,dr$$ where we have used that $G_N^{(D+1)}=G_N^{(d+2)}[R^{D-1-d} \,\textrm{Vol}(\mathcal{M}_{\textrm{c}})]$. Notice that the compact part enters through Kaluza-Klein reduction in the Newton’s constant also in this case where a warping factor occurs between the compact and the non compact part [@Barbon:2008sr; @Barbon:2008ut].

Expansion of the finite term near the horizon {#section near horizon}
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In this section we study the finite term of the holographic entanglement entropy introduced in the previous section. In particular, we consider the leading term of its expansion as the turning point of the minimal surface approaches the horizon, which means that the width $L$ of the strip in the boundary becomes large. As examples, we analyze the charged black hole in $AdS_{d+2}$ (section \[section nh charged bh\]), the warped black hole of [@Herzog:2009gd] (section \[section warped bh\]) and the perturbation of the Lifshitz background found in [@Gubser:2009cg] within the context of the Abelian Higgs model of [@Gubser:2008px] (section \[section perturbed lifshitz\]).\
The finite term in the expansion for small UV cutoff $a$ is given by $\mathcal{A}_d(z_{\textrm{max}},0)$, defined in (\[calArea\]). In order to consider its expansion as the turning point $z_{\textrm{max}}$ of the minimal surface gets 