  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.
