  Whereas a static observer in de Sitter (dS) space detects thermal radiation
emitted by the horizon, the dS isometries impose that the radiation is in
equilibrium with the background. This implies that for the static observer to
find the information paradox, the background must be deformed to quasi-dS space
in which the dS isometries are spontaneously broken. We study the condition
that the information paradox arises in quasi-dS space with the monotonically
increasing horizon size which is used to describe the inflationary cosmology.
For this purpose, the dimensional reduction of three-dimensional dS space with
thermal radiation modelled by the JT gravity coupled to CFT is considered. We
argue that when the central charge monotonically increases in time, the
information paradox arises but the conditions for the existence of the island
become more restrictive. As the central charge can be interpreted as the number
of degrees of freedom, the absence of the island in quasi-dS space supports the
entropy argument for the dS swampland conjecture.
