General form of entropy on the horizon of the universe in entropic
  cosmology
  Entropic cosmology assumes several forms of entropy on the horizon of the
universe, where the entropy can be considered to behave as if it were related
to the exchange (the transfer) of energy. To discuss this exchangeability, the
consistency of the two continuity equations obtained from two different methods
is examined, focusing on a homogeneous, isotropic, spatially flat, and
matter-dominated universe. The first continuity equation is derived from the
first law of thermodynamics, whereas the second equation is from the Friedmann
and acceleration equations. To study the influence of forms of entropy on the
consistency, a phenomenological entropic-force model is examined, using a
general form of entropy proportional to the $n$-th power of the Hubble horizon.
In this formulation, the Bekenstein entropy (an area entropy), the
Tsallis--Cirto black-hole entropy (a volume entropy), and a quartic entropy are
represented by $n=2$, $3$, and $4$, respectively. The two continuity equations
for the present model are found to be consistent with each other, especially
when $n=2$, i.e., the Bekenstein entropy. The exchange of energy between the
bulk (the universe) and the boundary (the horizon of the universe) should be a
viable scenario consistent with the holographic principle.
