  We propose a new class of cosmological unified dark sector models called
"{\em Generalized Logotropic Models}". They depend on a free parameter $n$. The
original logotropic model [P.H. Chavanis, Eur. Phys. J. Plus {\bf 130}, 130
(2015)] is a special case of our generalized model corresponding to $n=1$. In
our scenario, the Universe is filled with a single fluid, a generalized
logotropic dark fluid (GLDF), whose pressure $P$ includes higher order
logarithmic terms of the rest-mass density $\rho_m$. The total energy density
$\epsilon$ is the sum of the rest-mass energy density $\rho_m c^2$ and the
internal energy density $u$ which play the role of dark matter energy density
$\epsilon_m$ and dark energy density $\epsilon_{de}$, respectively. We
investigate the cosmological behavior of the generalized logotropic models by
focusing on the evolution of the energy density, scale factor, equation of
state parameter, decceleration parameter and squared speed of sound. Low values
of $n\le 3$ are favored. We also study the asymptotic behavior of the
generalized logotropic models. In particular, we show that the model presents a
phantom behavior and has three distinct ways of evolution depending on the
value of $n$. For $n\le 2$, it leads to a little rip and for $n>2$ to a big
rip. We predict the value of the big rip time as a function of $n$ without any
free (undetermined) parameter.
