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.
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