  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.
