Thermodynamics of interacting holographic dark energy with the apparent horizon as an IR cutoff

As soon as an interaction between holographic dark energy and dark matter is taken into account, the identification of IR cutoff with Hubble radius H^-1, in flat universe, can simultaneously drive accelerated expansion and solve the coincidence problem. Based on this, we demonstrate that in a non-flat universe the natural choice for IR cutoff could be the apparent horizon radius, rA=1/H²+k/a². We show that any interaction of dark matter with holographic dark energy, whose infrared cutoff is set by the apparent horizon radius, implies an accelerated expansion and a constant ratio of the energy densities of both components thus solving the coincidence problem. We also verify that for a universe filled with dark energy and dark matter the Friedmann equation can be written in the form of the modified first law of thermodynamics, dE=TₕdSₕ+WdV, at apparent horizon. In addition, the generalized second law of thermodynamics is fulfilled in a region enclosed by the apparent horizon. These results hold regardless of the specific form of dark energy and interaction term. Our study might reveal that in an accelerating universe with spatial curvature, the apparent horizon is a physical boundary from the thermodynamical point of view.