Thermodynamics of the inhomogeneous perfect fluid LTB model: Modified   Bekenstein-Hawking system
  The present work deals with three alternative generalized Bekenstein-Hawking formulation of thermodynamical parameters namely entropy and temperature for the universal thermodynamical system bounded by a horizon in the frame work of inhomogeneous perfect fluid Lemaitre-Tolman-Bondi (LTB) model of the Universe. For the first choice, the first law of thermodynamics holds only for the trivial de Sitter case of the LTB model while we need restriction on the evolution of the horizon radius for the validity of the generalized second law of thermodynamics. However, for the other two choices, the first law of thermodynamics holds with some specific (integral) form of the parameters involved but for generalized LTB models. 
Subhajit Saha and Subenoy Chakraborty
physics.gen-ph gr-qc