with the solution where and P is taken as a free parameter. We can assign any real value to P . For the results presented in this work, we used a negative value. This negative sign is necessarily to overcome the sign from the integral limits. The geometric and thermodynamic quantities of the Universe read 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 ther- modynamic, gravitational and cosmological quantities q ( t ) = [ ρ ( t )+3 p ( t )+3Π( t )] / 2 ρ ( t ) [33]. Besides the approximation in g ( z ), previous solution apparently depends on the free parameter P . In this section, we suggest a particular solution to overcome P . Eq. (12) 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 8 H = 4 9 α − $^{1}$γ 2 − γ . (29) IV. DE SITTER UNIVERSE H ( t ) = B exp( − Bt/ P ) − A (22) A = − 3[1 + (1 − r ) γ ] 2(1 − r ) , B = 1 αr , (23) a ( t ) = a$_{0}$ ( exp( − Bt/ P ) exp( − Bt/ P ) − A ) P /A , (24) ρ ( t ) = 3 H 2 = 3 ( B exp( − Bt/ P ) − A ) 2 , (25) T ( t ) = βρ r = β 3 B 2 [exp( − Bt/ P ) − A ] 2 ) r , (26) Π( t ) = − 2 ˙ H − 3 γH 2 = − B 2 P ( 2 exp( − Bt/ P ) + 3 γ P [exp( Bt/ ) − A ] 2 ) , (27) q ( t ) = d dt H − 1 − 1 = − 1 P exp( − Bt/ P ) − 1 . (28)