  In this paper we investigate the thermodynamics of the inner horizon and its
implication on the holographic description of the black hole. We focus on the
black holes with two physical horizons. Under reasonable assumption, we prove
that the first law of thermodynamics of the outer horizon always indicates that
of the inner horizon. As a result, the fact that the area product being
mass-independent is equivalent to the relation $T_+S_+=T_-S_-$, with $T_\pm$
and $S_\pm$ being the Hawking temperatures and the entropies of the outer and
inner horizon respectively. We find that the mass-independence of area product
breaks down in general Myers-Perry black holes with spacetime dimension
$d\geq6$ and Kerr-AdS black holes with $d\geq4$.
  Moreover we discuss the implication of the first laws of the outer and inner
horizons on the thermodynamics of the right- and left-moving sectors of dual
CFT in Kerr/CFT correspondence. We show that once the relation $T_+S_+=T_-S_-$
is satisfied, the central charges of two sectors must be same. Furthermore from
the thermodynamics relations, we read the dimensionless temperatures of
microscopic CFT, which are in exact agreement with the ones obtained from
hidden conformal symmetry in the low frequency scattering off the black holes,
and then determine the central charges. This method works well in well-known
cases in Kerr/CFT correspondence, and reproduce successfully the holographic
pictures for 4D Kerr-Newman and 5D Kerr black holes. We go on to predict the
central charges and temperatures of a possible holographic CFT description dual
to 5D doubly rotating black ring.
