Neumann Data and Second Variation Formula of Renormalized Area for Conformally Compact Static Spaces

In this paper, we derive the first and second variation formulas for the renormalized area for static Einstein spaces along a specific direction, demonstrating that the negativity of the Neumann data implies instability. Consequently, we obtain a rigidity result for the case when the conformal boundary is a warped product torus, which strengthens the result presented in \cite{GSW}.