  We use quantum extremal island method to study the information paradox on
certain cosmological setups known as three dimensional Kerr-de Sitter
spacetimes. To do so, we couple an auxiliary flat bath system to this spacetime
in timelike singularity and measure entropy of hawking radiation in its
asymptotic regions where the gravity is weak. We show that adding island
regions to the entanglement wedge of radiation causes its entropy to obey Page
curve. The boundary of island i.e. quantum extremal surface is located outside
the cosmological horizon in the region connected directly to the bath. Taking
the flat-space limit from the location of island and its related calculation in
the Kerr-de Sitter sapcetime results in the flat-space cosmology (FSC) island,
scrambling time and also Page time that were obtained in our previous paper. We
repeat the same calculation for the pure de Sitter spacetime and show that our
setup which neglects the effect of backreaction, leads also to a quantum
extremal surface outside the cosmological horizon. We calculate the scrambling
time and confirm the idea that pure de Sitter spacetime is a fast scrambler.
