Canonical energy and hairy AdS black holes

We propose the modified version of the canonical energy which was introduced originally by Hollands and Wald. Our construction depends only on the Euler-Lagrange expression of the system and thus is independent of the ambiguity in the Lagrangian. After some comments on our construction, we briefly mention on the relevance of our construction to the boundary information metric in the context of the AdS/CFT correspondence. We also study the stability of three-dimensional hairy extremal black holes by using our construction.