P}}\exp(-Bt/{\cal P})-1. \label{approx-q}\end{aligned}$$ $a_0$ is an arbitrary constant of the integration. The sign of $q$ indicates whether the Universe decelerates (positive) or accelerates (negative). $q$ can also be given as a function of the thermodynamic, gravitational and cosmological quantities $q(t)=[\rho(t)
  +3p(t)+3\Pi(t)]/2\rho(t)$ [@kolbBook].

de Sitter Universe  {#part1}
===================

Besides the approximation in $g(z)$, previous solution apparently depends on the free parameter ${\cal P}$. In this section, we suggest a particular solution to overcome ${\cal P}$. Eq. (\[init\]) can easily be obtained by assuming that $H$ doesn’t depend one $t$, i.e, de Sitter Universe. With a simple calculation, we get an estimation for $H$ $$\label{partcH}
H=\frac{4}{9}\frac{\alpha ^{-1}\gamma}{2-\gamma }.$$ The geometric and thermodynamic parameters of the Universe are given by $$\begin{aligned}
a(t) &=& a_{0}\exp \left[ \frac{4\alpha ^{-1}\gamma }{9(2-\gamma )}t\right], \label{partca} \\
\rho(t) &=& 3\left[ \frac{4\alpha ^{-1}\gamma }{9(2-\gamma )}\right] ^{2}, \label{partcrho}\\
T(t) &=& 3^{r}\beta \left[ \frac{4\alpha ^{-1}\gamma }{9(2-\gamma )}\right]
^{2r}, \label{partcT}\\
\Pi(t) &=& -3\gamma \left[ \frac{4\alpha
^{-1}\gamma }{9(2-\gamma )}\right] ^{2}, \label{partcPi}\\
q(t) &=& -1. \label{partcq}\end{aligned}$$ Although we have assumed here that the cosmic background is filled with viscous matter, the assumption that $H=const$ results in an exponential scale parameter, Eq. (\[partca\]). This behavior characterizes the de Sitter space, when $\Lambda=k=0$. $\rho$ and $T$ are 