  A non-monotonic behavior of the velocity gradient of a test particle
revolving around a rapidly rotating black hole in the locally non-rotating
frame of reference is known as the Aschenbach effect. This effect can serve as
a distinguishing signature of rapidly rotating black holes, being potentially
useful for the measurements of the astrophysical black hole spins. This paper
is the generalization of our previous research to the motion of spinning
particles around a rotating black hole with non-zero cosmological constant. We
show that both the particle's spin $s$ and the cosmological constant $\Lambda$
modify the critical value of the black hole spin $a_c$, for which the
Aschenbach effect can be observed; $a_c$ can increase or decrease depending on
the signs of $s$ and $\Lambda$. We also found that the particle's spin $s$ can
mimic the effect of the cosmological constant $\Lambda$ for a given $a_c$,
causing thus a discrepancy in the measurements of $s$, $\Lambda$ and $a_c$ in
the Aschenbach effect.
