  Here the Weyl curvature hypothesis is examined using the gravitational
entropy (GE). We have considered the family of C-metric accelerating black
holes and evaluated their corresponding gravitational entropy. Then we studied
the GE in some isotropic and anisotropic cosmologies utilizing the definition
proposed by Clifton, Ellis, and Tavakol, where the Bel-Robinson tensor is used
to determine the energy-momentum tensor of the free gravitational field. We
checked whether, in the vicinity of the initial cosmic singularity, the ratio
of the energy density of free gravity to that of matter density goes to zero or
not. We showed that whenever this is true, the gravitational entropy increases
monotonically with the structure formation of the universe and discussed the
conditions of validity for the Weyl curvature hypothesis. Subsequently, the
next part of the thesis deals with the validity of two different proposals of
gravitational entropy (GE) in traversable wormhole systems. We found that the
GE proposals do provide us with a consistent measure of GE in several wormhole
solutions. In the later portion of the thesis, we examined the validity of the
generalized second law of thermodynamics (GSLT) in an expanding FRW universe
filled with different variants of the Chaplygin gas. Lastly, we studied the
evolution of the FRW universe in the presence of variable modified Chaplygin
gas and obtained its temperature and other parameters as a function of the
redshift. Finally, the thesis is concluded.
