The accelerated expansion of the Universe can be interpreted as a tendency to satisfy holographic equipartition. It can be expressed by a simple law, $\mathrm{\Delta }V=\mathrm{\Delta }t(N_{\mathrm{surf}}-\epsilon N_{\text{bulk}})$, where $V$ is the Hubble volume in Planck units, $t$ is the cosmic time in Planck units, and $N_{\mathrm{surf}/\text{bulk}}$ is the number of degrees of freedom on the horizon/bulk of the Universe. We show that this holographic equipartition law effectively implies the maximization of entropy. In the cosmological context, a system that obeys the holographic equipartition law behaves as an ordinary macroscopic system that proceeds to an equilibrium state of maximum entropy. We consider the standard $\mathrm{\Lambda }\mathrm{CDM}$ model of the Universe and show that it is consistent with the holographic equipartition law. Analyzing the entropy evolution, we find that it also proceeds to an equilibrium state of maximum entropy.

The accelerated expansion of the Universe can be interpreted as a quest for satisfying holographic equipartition. It can be expressed by a simple law, $\mathrm{\Delta }V=\mathrm{\Delta }t(N_{\mathrm{surf}}-N_{\mathrm{bulk}})$ which leads to the standard Friedmann equation. This novel idea suggested by Padmanabhan in the context of general relativity has been generalized by Cai and Yang et al. to Gauss-Bonnet and Lovelock gravities for a spatially flat universe in different methods. We investigate the consistency of these generalizations with the constraints imposed by the maximum entropy principle. Interestingly, both these generalizations imply entropy maximization even if their basic assumptions are different. Further, we analyze the consistency of Verlinde’s emergent gravity with the maximum entropy principle in the cosmological context. In particular, we consider the generalization suggested by Shu and Gong, in which an energy flux through the horizon is assumed, in addition. Even though the conceptual formulations are different, these two emergent perspectives of gravity describes a universe which behaves as an ordinary macroscopic system. Our results provide further support to the emergent gravity paradigm.

The emergence of cosmic space as cosmic time progresses is an exciting idea advanced by Padmanabhan to explain the accelerated expansion of the universe. The generalization of Padmanabhan's conjecture to the non-flat universe has resulted in scepticism about the choice of volume such that the law of emergence can not be appropriately formulated if one uses proper invariant volume. The deep connection between the first law of thermodynamics and the law of emergence [1], motivate us to explore the status of the first law in a non-flat universe when one uses proper invariant volume. We have shown that the first law of thermodynamics, dE = TdS + WdV cannot be formulated properly for a non-flat universe using proper invariant volume. We have also investigated the status of the first law of the form -dE = TdS in a non-flat universe. We have shown that the energy change dE(nu) within the horizon and the outward energy flux are not equivalent to each other in a non-flat universe when we use the proper invariant volume. We further point out that the consistency between the above two forms of the first law claimed in ref. [2] will hold only with the use of the areal volume of the horizon. The failure in formulating the first law of thermodynamics with the use of invariant volume shows that the invariant volume will be a poor choice to describe any thermodynamic process in cosmology.

According to the principle of emergence, the expansion of the Universe can be explained as the emergence of space with the progress of cosmic time. We have analytically solved the equation of emergence proposed by Padmanabhan by assuming the Komar energy density ρ + 3P as a function of the Hubble parameter. The resulting model describes the evolution of the Universe, which proceeds towards a final de Sitter state. Model parameters have been extracted using the cosmological observational data. Further, the horizon entropy evolution of the model has been studied. The model predicts a Universe having a transition from a prior decelerated epoch to a late accelerated epoch and reasonably predicts the cosmological constant.