5 [ 3 a T$_{sc}$ ( a ) ω$_{sc}$ ( a ) + T ′ $_{sc}$( a ) ] ( ω$_{sc}$ ( a ) + 1) = T$_{sc}$ ( a ) ω ′ $_{sc}$( a ) . (37) T$_{sc}$ ( a ) = T$_{0}$ ( ω$_{sc}$ ( a ) + 1 ω$_{0}$ + 1 )( 1 a 3 ω$_{sc}$ ( a ) ) × exp [ − 3 ∫ 1 a daω ′ $_{sc}$( a ) ln a ] , (38) ˙ s$_{sc}$ + 3 Hs$_{sc}$ = 0 , (33) ˙ n + 3 Hn = 0 . (34) n ( a ) = n$_{0}$ ( a a$_{0}$ ) 3 , s$_{sc}$ ( a ) = s$_{0}$ ( a a$_{0}$ ) 3 . (35) B. The thermodynamical variables of dark energy fluid form as Eq.(32), when one considers the effective LQC with holonomy corrections [16]. This difference is reason- able. Because we consider the inverse volume correction in the semiclassical regions, but the authors of [16] con- sidered the holonomy correction. Notice that there are many differences between inverse volume correction and holonomy correction. For example, the inverse volume correction modifies the Klein-Gorden equation [34], but holonomy correction does not [35]; and the semiclassical energy density and effective pressure which caused by the inverse volume correction are different from the ones caused by the holonomy correction [36]. T$_{sc}$ ( a ) [ ∂ω$_{sc}$ ( a ) ρ$_{sc}$ ∂T$_{sc}$ ( a ) ] n = ( ω$_{sc}$ ( a ) + 1) ρ$_{sc}$ − n [ ∂ρ$_{sc}$ ( a ) ∂n ] T$_{sc}$ . (36) Considering ˙ ρ$_{sc}$ ( a ) = ˙ n ( ∂ρ$_{sc}$ ( a ) ∂n ) T$_{sc}$ ( a $_{)}$+ ˙ T$_{sc}$ ( a ) ( ∂ρ$_{sc}$ ( a ) ∂T$_{sc}$ ( a ) ) n and Eqs.(10) and (36), we can get the evolution of mod- ified temperature with respect to the semiclassical EoS parameters prefect fluid can be described by Just as the mention in Sec.II, the modified EoS param- eters ω$_{sc}$ ( a ) are possible -1 crossing. It is obvious that the modified temperature which is described by above equation is zero when ω$_{sc}$ ( a ) = − 1 for ω ′ $_{sc}$( a ) = 0. In- tegrating the above equation when ω$_{sc}$ ( a ) = − 1, we can get the expression for T$_{sc}$ ( a ) In this section, we will discuss the thermodynamics properties of the dark energy in the inner side of the ap- parent horizon. Some authors advised that the tempera- ture of dark energy fluid has the same temperature of the apparent horizon which the temperature is described by Hawking temperature, but this may be get a ill-defined entropy when ω$_{cl}$ < − 1 [24]. And other authors find the dark energy either have negative entropy or temperature in the classical regions [21–23]. In this subsection, we will consider the temperature and entropy of the dark energy in the inner side of the apparent horizon. And we will assume the dark energy is a ideal fluid and with the chemical potential. This subsection, we will follow the method of [20] to discuss the temperature and entropy of the dark energy in semiclassical LQC scenario. The entropy fluid and particle fluid are described by S µ sc = s$_{sc}$u µ and N µ = nu $^{µ}$, with the particle number density n and the entropy density s$_{sc}$ , respectively. Con- sidering they satisfy S µ ; = 0 , N µ ; = 0, it is easy to obtain with the present temperature T$_{0}$ . We have considered that the present scale factor a$_{0}$ = 1. There are four points should be noticed. First, if ω$_{0}$ = − 1 and T$_{0}$ = 0, T$_{sc}$ ( a ) will be divergent. But if we consider ω$_{sc}$ ( a ) de- scribes today’s dark energy, ω$_{sc}$ ( a ) should not be modi- fied by the quantum effect for a$_{0}$ ≫ a$_{⋆}$ and is substituted by ω$_{cl}$ ( a ). In this case, Eq.(37) describes the temper- ature for the dark energy in the classical regions. The temperature of dark energy with ω ( a$_{0}$ ) = − 1 should be zero, for ω ′ $_{cl}$( a ) | a = a$_{0}$ = 0 in Eq.(37). So, T$_{0}$ = 0 when ω$_{0}$ = − 1 if the EoS ω$_{cl}$ is a function of the scale fac- tor. Second, the temperature is zero when ω$_{sc}$ ( a ) = − 1. But this ”-1” EoS parameter is caused by the effective behavior. Third, for T$_{0}$ and ω$_{0}$ + 1 have the same sign, if ω$_{sc}$ ( a ) > − 1 the temperature is positive, the tempera- ture is negative when ω$_{sc}$ ( a ) < − 1, and T$_{sc}$ ( a ) = 0 when ω$_{sc}$ ( a ) = − 1, this is as same as the behavior of temper- ature in the classical regions [20]. But notice that ω$_{sc}$ are modified EoS parameters and includes the quantum geometry effect. Fourth, the scale factor is in the regions a$_{i}$ < a < a$_{⋆}$ , so 1 a in Eq.(38) will not divergent, this is different from [20] in which the authors considered the thermodynamical properties of dark energy in the classi- cal regions. The solutions for above equations are Now we consider the thermodynamics. We assume the thermodynamic properties is described by the particle number density n and the temperature T$_{sc}$ . We have assumed the quantum effect will modify the tempera- ture. This is reasonable. If assuming that the modified energy density and the temperature is still satisfy the Stenfan-Boltzmann law ρ$_{sc}$ ∝ f ( T$_{sc}$ ), the temperature T$_{sc}$ should be dependent on a , for the energy density is the function of the scale factor a , and then the temper- ature should be modified by the semiclassical quantum effect. We will see that the Stenfan-Boltzmann law still is tenable in our model later on. We consider the Gibbs law, T$_{sc}$ ( ∂P$_{sc}$ ∂T$_{sc}$ ) n = ρ$_{sc}$ + P$_{sc}$ − n [ ∂ρ$_{sc}$ ( a ) ∂n ] $_{T$_{sc}$}$, then the Now we have the expressions for modified temperature and modified energy density. It is easy to obtain the