How appropriate are the gravitational entropy proposals for traversable wormholes

In this paper we have examined the validity of some proposed definitions of gravitational entropy in the context of traversable wormhole solutions of the Einstein field equations. Here we have adopted two different kinds of proposals of gravitational entropy and compared them in the context of traversable wormholes. The first one is the phenomenological approach proposed in Rudjord et al \cite{entropy1} and expanded by Romero et al in \cite{entropy2}, which is a purely geometric method of measuring gravitational entropy. The later one is the Clifton-Ellis-Tavakol (CET) proposal \cite{CET} for the gravitational entropy which is based on relativistic thermodynamics. Considering some of the Lorentzian traversable wormholes along with the Brill solution for NUT wormholes and the AdS wormholes, we have evaluated the gravitational entropy density for these systems. The application of CET proposal is limited to Petrov D class spacetimes whereas the geometric method can be used in almost every kind of spacetime. We end up by discussing the merits of such an analysis and the demerits in the particular case of axisymmetric wormholes and comment on the possible resolution of the problem.