Different faces of generalized holographic dark energy

In the formalism of generalized holographic dark energy (HDE), the holographic cut-off is generalized to depend upon $L_\mathrm{IR} = L_\mathrm{IR} \left( L_\mathrm{p}, \dot L_\mathrm{p}, \ddot L_\mathrm{p}, \cdots, L_\mathrm{f}, \dot L_\mathrm{f}, \cdots, a\right)$ with $L_\mathrm{p}$ and $L_\mathrm{f}$ are the particle horizon and the future horizon, respectively (moreover $a$ is the scale factor of the universe). Based on such formalism, in the present paper, we show that a wide class of dark energy (DE) models can be regarded as different candidates of the generalized HDE family, with respective cut-offs. This can be thought as a symmetry between the generalized HDE and different DE models. In this regard, we consider several entropic dark energy models - like Tsallis entropic DE, the R\'{e}nyi entropic DE, and the Sharma-Mittal entropic DE - and showed that they are indeed equivalent with the generalized HDE. Such equivalence between the entropic DE and the generalized HDE is extended to the scenario where the respective exponents of the entropy functions are allowed to vary with the expansion of the universe. Besides the entropic DE models, the correspondence with the generalized HDE is also established for the Quintessence and for the Ricci DE models. In all the above cases, the effective equation of state (EoS) parameter corresponds to the holographic energy density are determined, by which the equivalence of various DE models with the respective generalized HDE models are further confirmed. The equivalent holographic cut-offs are determined by two ways: (1) in terms of the particle horizon and its derivatives, (2) in terms of the future horizon horizon and its derivatives.