  We investigate black hole thermodynamics involving a scalar hair which is
dual to a momentum relaxation of the dual field theory. This black hole
geometry is able to be classified by two parameters. One is a momentum
relaxation and the other is a mass density of another matter localized at the
center. Even though all parameters are continuous, there exists a specific
point where its thermodynamic interpretation is not continuously connected to
the one defined in the other parameter regime. The similar feature also appears
in a topological AdS black hole. In this work, we show why such an unusual
thermodynamic feature happens and provide a unified way to understand such an
exotic black hole thermodynamically in the entire parameter range.

High Energy Physics - Theory;General Relativity and Quantum Cosmology