  We prove R\'enyi entropic inequalities in a holographic setup based on the
recent proposal for the holographic formula of R\'enyi entropies when the bulk
is stable against any perturbation. Regarding the R\'enyi parameter as an
inverse temperature, we reformulate the entropies in analogy with statistical
mechanics, which provides us a concise interpretation of the inequalities as
the positivities of entropy, energy and heat capacity. This analogy also makes
clear a thermodynamic structure in deriving the holographic formula. As a
by-product of the proof we obtain a holographic formula to calculate the
quantum fluctuation of the modular Hamiltonian. A few examples of the capacity
of entanglement are examined in detail.
