The "universal property" of Horizon Entropy Sum of Black Holes in Four Dimensional Asymptotical (anti-)de-Sitter Spacetime Background

We present a new ``universal property'' of entropy, that is the ``entropy sum'' relation of black holes in four dimensional (anti-)de-Sitter asymptotical background. They depend only on the cosmological constant with the necessary effect of the un-physical ``virtual'' horizon included in the spacetime where only the cosmological constant, mass of black hole, rotation parameter and Maxwell field exist. When there is more extra matter field in the spacetime, one will find the ``entropy sum'' is also dependent of the strength of these extra matter field. For both cases, we conclude that the ``entropy sum'' does not depend on the conserved charges $M$, $Q$ and $J$, while it does depend on the property of background spacetime. We will mainly test the ``entropy sum'' relation in static, stationary black hole and some black hole with extra matter source (scalar hair and higher curvature) in the asymptotical (anti-)de-sitter spacetime background. Besides, we point out a newly found counter example of the mass independence of the ''entropy product'' relation in the spacetime with extra scalar hair case, while the ``entropy sum'' relation still holds. These result are indeed suggestive to some underlying microscopic mechanism. Moreover, the cosmological constant and extra matter field dependence of the ``entropy sum'' of all horizon seems to reveal that ``entropy sum'' is more general as it is only related to the background field. For the case of asymptotical flat spacetime without any matter source, we give a note for the Kerr black hole case in appendix. One will find only mass dependence of ``entropy sum'' appears. It makes us believe that, considering the dependence of ``entropy sum'', the mass background field may be regarded as the next order of cosmological constant background field and extra matter field.Comment: 14 pages, no figures, JHEP forma

Thermodynamic relations for entropy and temperature of multi-horizons black holes

We present some entropy and temperature relations of multi-horizons, even including the "virtual" horizon. These relations are related to product, division and sum of entropy and temperature of multi-horizons. We obtain the additional thermodynamic relations of both static and rotating black holes in three and four dimensional (A)dS spacetime. Especially, a new dimensionless, charges-independence and $T_+S_+=T_-S_-$ like relation is presented. This relation does not depend on the mass, electric charge, angular momentum and cosmological constant, as it is always a constant. These relations lead us to get some interesting thermodynamic bound of entropy and temperature, including the Penrose inequality which is the first geometrical inequality of black holes. Besides, based on these new relations, one can obtain the first law of thermodynamics and Smarr relation for all horizons of black hole.Comment: 12 pages, no figures, title changed, references adde