  In this paper, we investigate the structure of neutron stars by considering
both the effects of the cosmological constant and the existence of quark matter
for neutron stars in Einstein's gravity. For this purpose, we use a suitable
equation of state (EoS) which includes a layer of hadronic matter, a mixed
phase of quarks and hadrons, and a quark matter in the core. To investigate the
effect of the cosmological constant on the structure of hybrid neutron stars,
we utilize the modified TOV equation in Einstein-$\Lambda $ gravity. Then we
drive the mass-radius relation for different values of the cosmological
constant. Our results show that for small values of the cosmological constant
($\Lambda $), especially for the cosmological constant from the cosmological
perspective $(\Lambda =10^{-52}$ $m^{-2})$, $\Lambda $ has no significant
effect on the structure of hybrid neutron stars. But for higher values, for
example, by considering $\Lambda >10^{-14}$ $m^{-2}$, this quantity affects the
maximum mass and radius of these stars. We find an upper limit for the
cosmological constant as $\Lambda <9\times 10^{-13}m^{-2}$, based on the fact
that the gravitational redshift cannot be more than $1$ for stars. The maximum
mass and radius of these stars decrease by increasing the cosmological constant
$\Lambda$. Also, by determining and analyzing radius, the compactness,
Kretschmann scalar, and gravitational red shift of the hybrid neutron stars
with $M=1.4M_{\,\odot }$ in the presence of the cosmological constant, we find
that by increasing $\Lambda$, they are contracted. Also, our results for
dynamical stability show that these stars satisfy this condition.
