  We note that the observable part of universe at a certain time t_P is
necessarily limited, when there is a beginning of universe. We argue that an
appropriate spacetime region associated with an observer from tI to t_P is the
causal diamond which is the overlap of the past/future of the observer at
t_P/t_I respectively. We also note that the overlap surface \partial D of the
future and the past lightcones bisects the spatial section including \partial D
into two regions D and \bar D where D is the region inside the causal diamond
and \bar D the remaining part of the spatial section. We propose here that the
entropy of universe associated with a causal diamond is given by an
entanglement entropy where one is tracing over the Hilbert space associated
with the region \bar D which is not accessible by the observer. We test our
proposal for various examples of cosmological spacetimes, including flat or
open FRW universes, by showing that the entropy as the area of \partial D
divided by 4G is a non-decreasing function of time t_P as dictated by the
generalized second law of thermodynamics. The closed, recollapsing universe
corresponds to a finite system and there is no reason to expect the validity of
the generalized second law for such a finite system.
