Loop quantum cosmology (LQC) is very powerful to deal with the behavior of early universe. Moreover, the effective loop quantum cosmology gives a successful description of the universe in the semiclassical region. We consider the apparent horizon of the Friedmann-Robertson-Walker universe as a thermodynamical system and investigate the thermodynamics of LQC in the semiclassical region. The effective density and effective pressure in the modified Friedmann equation from LQC not only determine the evolution of the universe in LQC scenario but also are actually found to be the thermodynamic quantities. This result comes from the energy definition in cosmology (the Misner-Sharp gravitational energy) and is consistent with thermodynamic laws. We prove that within the framework of loop quantum cosmology, the elementary equation of equilibrium thermodynamics is still valid.

Loop quantum gravity (LQG) [

In LQC, the phase space for spatially flat universe is
spanned by coordinates

In recent years, many authors endeavor to study the
thermodynamics [

To resolve this issue, we develop a procedure to study
the thermodynamic properties at the apparent horizon of the
Friedmann-Robertson-Walker (FRW) universe. Our analysis is based on the
effective theory of LQC and the homogeneous and isotropic cosmological
settings. Fundamentally, comparing the modified
Friedmann equation with the ordinary one, we derive the effective density and
pressure of the perfect fluid. Then, we introduce the Misner-Sharp energy
[

This paper is organized as follows. In Section

In this section, we give a short review of the
effective framework of LQC before we study the thermodynamics. The classical
form of Hamiltonian for spatially flat universe is

Let us begin with the effective LQC description of the
universe evolution. For a spatially homogenous and isotropic universe described
by the FRW metric, the line element is represented by

For the FRW universe, the dynamical apparent
horizon, (without the whole evolution history of the universe, one
cannot know whether there is a cosmological event horizon.
However, apparent horizon always exists in the FRW
universe since it is a local quantity of spacetime.) defined as the sphere with vanishing
expansion [

We now introduce the Misner-Sharp spherically
symmetric gravitational energy

In terms of the apparent horizon radius (

To examine the fundamental relation of thermodynamics
in the setup of LQC, we consider the apparent horizon of the FRW universe as a
thermodynamical system. An ansatz is made. Assume that the apparent horizon has
an associated Hawking temperature

Taking derivative of the energy equation (

In Conclusion, we have investigated the thermodynamic
properties of the universe in LQC scenario and found that the fundamental
relation of thermodynamics is valid in the effective LQC scenario. We found
that the effective density

In the following, we briefly comment on the physical
meanings from the expressions of the effective energy density and pressure.
When the energy density is much smaller than the quantum critical density (

The work was supported by the National Natural Science Foundation of China (no. 10875012). The first author is indebted to Dr. Dah-Wei Chiou for his helpful discussions.