Evolution of the universe in entropic cosmologies via different
  formulations
  We study two types of entropic-force models in a homogeneous, isotropic,
spatially flat, matter-dominated universe. The first type is a `$\Lambda(t)$
type' similar to $\Lambda(t)$CDM (varying-lambda cold dark matter) models in
which both the Friedmann equation and the acceleration equation include an
extra driving term. The second type is a `BV type' similar to bulk viscous
models in which the acceleration equation includes an extra driving term
whereas the Friedmann equation does not. In order to examine the two types
systematically, we consider an extended entropic-force model that includes a
Hubble parameter ($H$) term and a constant term in entropic-force terms. The
$H$ term is derived from a volume entropy whereas the constant term is derived
from an entropy proportional to the square of an area entropy. Based on the
extended entropic-force model, we examine four models obtained from combining
the $H$ and constant terms with the $\Lambda(t)$ and BV types. The four models
agree well with the observed supernova data and describe the background
evolution of the late universe properly. However, the evolution of first-order
density perturbations is different in each of the four models, especially for
low redshift, assuming that an effective sound speed is negligible. The
$\Lambda(t)$ type is found to be consistent with the observed growth rate of
clustering, in contrast with the BV type examined in this study. A unified
formulation is proposed as well, in order to examine density perturbations of
the two types systematically.
