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Our Universe is expected to finally approach a de Sitter universe whosehorizon is considered to be in thermal equilibrium. In the present article,both the energy stored on the horizon and its thermodynamic fluctuations areexamined through the holographic equipartition law. First, it is confirmed thata flat Friedmann--Robertson--Walker universe approaches a de Sitter universe,using a cosmological model close to lambda cold dark matter ($\Lambda$CDM)models. Then, based on the holographic equipartition law, the energy density ofthe Hubble volume is calculated from the energy on the Hubble horizon of a deSitter universe. The energy density for a de Sitter universe is constant andthe order of the energy density is consistent with the order of that for theobserved cosmological constant. Second, thermodynamic fluctuations of energy onthe horizon are examined, assuming stable fluctuations around thermalequilibrium states. A standard formulation of the fluctuations for a canonicalensemble is applied to the Hubble horizon of a de Sitter universe. Thethermodynamic fluctuations of the energy are found to be a universal constantcorresponding to the Planck energy, regardless of the Hubble parameter. Incontrast, the relative fluctuations of the energy can be characterized by theratio of the one-degree-of-freedom energy to the Planck energy. At the presenttime, the order of the relative fluctuations should be within the range of adiscrepancy derived from a discussion of the cosmological constant problem,namely a range approximately from $10^{-60}$ to $10^{-123}$. The presentresults may imply that the energy stored on the Hubble horizon is related to akind of effective dark energy, whereas the energy that can be `maximally'stored on the horizon may behave as if it were a kind of effective vacuum-likeenergy in an extended holographic equipartition law.

Energy stored on a cosmological horizon and its thermodynamic fluctuations in holographic equipartition law