content_image/395431.jpg content_image/395435.jpg content_image/395436.jpg content_image/395439.jpg content_image/395437.jpg content_image/395438.jpg content_image/395430.jpg content_image/395433.jpg content_image/395434.jpg content_image/395432.jpg
Integrating the above equation, it is easy to get the so- lution for ρ$_{sc}$ 

<img src='content_image/395431.jpg'>

with ρ$_{0}$, a$_{0}$, ω$_{0}$ denote today’s energy density, scale factor and EoS parameter respectively. For a$_{0}$ ≫ a$_{⋆}$ , it’s not necessary to consider the quantum effect of those present values (For conservation, we will define the subscript ’0’ denotes the present value of a physical quantity.). And the prime means the derivation with respect to the scale factor a . 

In the above discussion, we defined the semiclassical energy density as ρ$_{sc}$ = E$_{m}$/a $^{3}$, but this is not the only definition. Singh has showed us that there are two ways to obtain the semiclassical density [27]. One way is just like the definition we discussed above. The other one is to define a density operator ˆ ρ$_{q}$ = ̂ H$_{m}$/a 3 and then take their eigenvalues. The eigenvalues for ˆ ρ$_{q}$ can be obtained by considering ˆ H$_{m}$ and 1 /a 3 [27] 

<img src='content_image/395435.jpg'>

in which d$_{j,l}$ = D$_{l}$ ( q ) a − 3 is the eigenvalue for ̂ 1 /a 3 for the large j . And D$_{l}$ is given as [27] 1 − 2 l/ 3 

<img src='content_image/395436.jpg'>

where q = a $^{2}$/a2 $_{⋆}$, and l is the quantum ambiguity pa- rameters with 0 < l < 1 [26]. For a ≲ a$_{⋆}$ , D$_{l}$ ≲ 1, this means ρ$_{sc}$ ≲ ρ$_{q}$ . 

Now, we turn to discuss the relationship between the semiclassical energy density and the classical one. The classical energy conservation is 

Considering the varying EoS parameter ω$_{cl}$ ( a ) = ρ$_{cl}$/P$_{cl}$ , the above equation can be written as 

It is easy to obtain the expression for ρ$_{cl}$ from the above equation 

<img src='content_image/395439.jpg'>

Considering the definition of ρ$_{q}$ , we can get the eigenval- ues for ̂ ρ$_{q}$ using the eigenvalues for ̂ 1 /a $^{3}$, d$_{j,l}$ , instead of 

<img src='content_image/395437.jpg'>

<img src='content_image/395438.jpg'>

a − 3 in Eq.(16). So, the semiclassical expression for ρ$_{q}$ can be written as 

<img src='content_image/395430.jpg'>

Remembering the relationship between ρ$_{q}$ and ρ$_{sc}$ , which is described by Eq.(12), we can get the relationship be- tween ρ$_{sc}$ and ρ$_{cl}$ 

Now we want to get the relationship between ω$_{sc}$ ( a ) and ω$_{cl}$ ( a ). Differentiating ln ρ$_{sc}$ with respect to ln a , one can obtain 

<img src='content_image/395433.jpg'>

Note that d ln ρ$_{sc}$ ( a ) d ln a = dρ$_{sc}$ ( a ) Hρ$_{sc}$ ( a ) dt = − 3[1 + ω$_{sc}$ ( a )], and d ln ρ$_{cl}$ ( a ) d ln a = dρ$_{cl}$ ( a ) Hρ$_{cl}$ ( a ) dt = − 3[1 + ω$_{cl}$ ( a )], it is easy to get 

<img src='content_image/395434.jpg'>

Remembering that ω$_{cl}$ ( a ) may be a negative value for dark energy fluid, so we need choose the absolute value when we calculate its logarithm, but it dose not change the answer. Considering Eq.(20), it is easy to find that the effective EoS in loop quantum cosmology is possible ”-1” crossing. If ω$_{cl}$ = const . , Eq.(20) will still hold, but the second term in the square brackets is zero. So the constant classical EoS will be a varying term in the effective region. 

In this section, we get the equation of the effective EoS of dark energy with a varying EoS in loop quantum cos- mology. Now, we can turn to consider the thermodynam- ical properties of dark energy in loop quantum cosmology scenario. 

## III. THERMODYNAMICS PROPERTIES OF DARK ENERGY 

In this section, we will discuss the thermodynamics properties of dark energy in semiclassical regions. This section includes two subsections. In the first subsection, we will discuss the entropy-area relation of apparent hori- zon in semiclassical scenario. In the next subsection, we will show the temperature, chemical potential and en- tropy of dark energy fluid inner of the apparent horizon. 

According to the fact that the Einstein equation can be rewritten as the form dE$_{m}$ = T dS + W$_{m}$dV [28–30], 

## A. Modified entropy-area relation 

<img src='content_image/395432.jpg'>

