Thermodynamical properties of interacting holographic dark energy model   with apparent horizon
  We have investigated the thermodynamical properties of the universe with dark energy. It is demonstrated that in a universe with spacial curvature the natural choice for IR cutoff could be the apparent horizon radius. We shown that any interaction of pressureless dark matter with holographic dark energy, whose infrared cutoff is set by the apparent horizon radius, implying a constant effective equation of state of dark component in a universe. In addition we found that for the static observer in space, the comoving distance has a faster expansion than the apparent horizon radius with any spatial curvature. We also verify that in some conditions the modified first law of thermodynamics could return to the classic form at apparent horizon for a universe filled with dark energy and dark matter. Besides, the generalized second law of thermodynamics is discussed in a region enclosed by the apparent horizon. 
Bin Liu, Xian-Ru Hu and Jian-Bo Deng
hep-th