  This study focuses on the impact of the cosmological constant on
hyperbolically symmetric matter configurations in static background. We extend
the work of Herrera \cite{herrera2021hyperbolically} and describe the
influences of such repulsive character on a few realistic features of
hyperbolical anisotropic fluids. After describing Einstein-$\Lambda$ equations
of motion, we elaborated the corresponding mass function along with its
conservation laws. In our study, besides observing negative energy density, we
notice the formation of Minkowskian core as matter content is compelled itself
not to follow inward motion near the axis of symmetry. Three families of
solution are found in $\Lambda$-dominated epoch. First is calculated by keeping
zero value to Weyl scalar, while the second solution maintains zero complexity
in the subsequent changes of the hyperbolical compact object. However, the last
model encompasses stiff fluid within the self-gravitating system. Such type of
theoretical setup suggests its direct link to study a few particular quantum
scenarios where negative behavior of energy density is noticed at $\Lambda$
dominated regime.
