Wormhole Thermodynamics at Apparent Horizons

In this paper, we discuss the thermodynamic properties of the evolving Lorentzian wormhole. For the shape function $b(r) = r_{0}^2/r$, it is shown that the wormhole spacetime admits two apparent horizons, the inner and the outer one. The inner horizon expands while the outer contracts with the passage of time. Corresponding to these horizons, we have three types of wormholes, regular, extreme and the naked wormholes. Moreover, it is shown that the Einstein field equations can be rewritten as a first law of thermodynamics $dE=TdS+WdV$, at the apparent horizons of the wormhole, where $E=\rho V$, $T = \kappa/2\pi$, $S=A/4G$, $W=(\rho-P)/2$ and $V = {4/3}\pi \tilde{r}_{A+}^3$ are the total matter energy, horizon temperature, wormhole entropy, work density and the volume of the wormhole respectively.