  The future space-borne detectors will provide the possibility to detect
gravitational waves emitted from extreme mass ratio inspirals of stellar-mass
compact objects into supermassive black holes. It is natural to expect that the
spin of the compact object and cosmological constant will affect the orbit of
the inspiral process and hence lead to the considerable phase shift of the
corresponding gravitational waves. In this paper, we investigate the motion of
a spinning test particle in the spinning black hole background with a
cosmological constant and give the order of motion deviation induced by the
particle's spin and the cosmological constant by considering the corresponding
innermost stable circular orbit. By taking the neutron star or kerr black hole
as the small body, the deviations of the innermost stable circular orbit
parameters induced by the particle's spin and cosmological constant are given.
Our results show that the deviation induced by particle's spin is much larger
than that induced by cosmological constant when the test particle locates not
very far away from the black hole, the accumulation of phase shift during the
inspiral from the cosmological constant can be ignored when compared to the one
induced by the particle's spin. However when the test particle locates very far
away from the black hole, the impact from the cosmological constant will
increase dramatically. Therefore the accumulation of phase shift for the whole
process of inspiral induced by the cosmological constant and the particle's
spin should be handled with caution.
