  In this paper, aimed at exploring the fundamental properties of isoperimetric
region in $3$-manifold $(M^3,g)$ which is asymptotic to Anti-de
Sitter-Schwarzschild manifold with scalar curvature $R\geq -6$, we prove that
connected isoperimetric region $\{D_i\}$ with $\mathcal{H}_g ^3(D_i)\geq
\delta_0>0$ cannot slide off to the infinity of $(M^3,g)$ provided that
$(M^3,g)$ is not isometric to the hyperbolic space. Furthermore, we prove that
isoperimetric region $\{D_i\}$ with topological sphere $\{\partial D_i\}$ as
boundary is exhausting regions of $M$ if Hawking mass $m_H(\partial D_i)$ has
uniform bound. In the case of exhausting isoperimetric region, we obtain a
formula on expansion of isoperimetric profile in terms of renormalized volume.
