geometry which is given by $f(z)=O(z_0-z)$ for the Schwarzschild and the non extremal case and by $f(z)=O\big((z_0-z)^2\big)$ for the extremal case (see the appendix \[app chargedBH\]). As a check, one can perform the expansion of the finite term just described substituting to the emblacking function $f(z)$ its near horizon approximation $f_{\textrm{nh}}(z)$ and verify that the same divergence shown above are obtained.\
We remark that for all the black holes we are considering the horizon is non compact; therefore the wrapping of the minimal surface around the horizon in the large $L$ limit described e.g. in [@Ryu:2006bv; @Ryu:2006ef; @Headrick:2007km; @Azeyanagi:2007bj] does not occur. In the appendix \[app more on nh\] we employ the splitting (\[split 1\]) of the finite term to study the $O(1)$ term in (\[expansion nh rn T0\]) and discuss the approximation obtained by using the near horizon geometry.

Warped black hole {#section warped bh}
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In this section we employ the observation just made about the role of the region close to the horizon and apply the expansion described in (\[eps def\]) and (\[A0 expanded\]) to a black hole where only the near horizon geometry is known.\
In [@Herzog:2009gd] a minimal consistent truncation of the type IIB supergravity 