Thermodynamic product formula for a Taub–NUT black hole

We derive various important thermodynamic relations of the inner and outer horizons in the background of the Taub–NUT (Newman–Unti–Tamburino) black hole in four-dimensional Lorentzian geometry. We compare these properties with the properties of the Reissner–Nordström black hole. We compute the area product, area sum, area subtraction, and area division of black hole horizons. We show that they all are not universal quantities. Based on these relations, we compute the area bound of all horizons. From the area bound, we derive an entropy bound and an irreducible mass bound for both horizons. We further study the stability of such black holes by computing the specific heat for both horizons. It is shown that due to the negative specific heat, the black hole is thermodynamically unstable. All these calculations might be helpful in understanding the nature of the black hole entropy (both interior and exterior) at the microscopic level.