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This study focuses on the impact of the cosmological constant onhyperbolically symmetric matter configurations in static background. We extendthe work of Herrera \cite{herrera2021hyperbolically} and describe theinfluences of such repulsive character on a few realistic features ofhyperbolical anisotropic fluids. After describing Einstein-$\Lambda$ equationsof motion, we elaborated the corresponding mass function along with itsconservation laws. In our study, besides observing negative energy density, wenotice the formation of Minkowskian core as matter content is compelled itselfnot to follow inward motion near the axis of symmetry. Three families ofsolution are found in $\Lambda$-dominated epoch. First is calculated by keepingzero value to Weyl scalar, while the second solution maintains zero complexityin the subsequent changes of the hyperbolical compact object. However, the lastmodel encompasses stiff fluid within the self-gravitating system. Such type oftheoretical setup suggests its direct link to study a few particular quantumscenarios where negative behavior of energy density is noticed at $\Lambda$dominated regime.

Spatially Hyperbolic Gravitating Sources in $\Lambda$-Dominated Era