  In this article, we investigate the quantum circuit complexity and
entanglement entropy in the recently studied black hole gas framework using the
two-mode squeezed states formalism written in arbitrary dimensional spatially
flat cosmological Friedmann-Lema$\hat{i}$tre-Robertson-Walker (FLRW) background
space-time. We compute the various complexity measures and study the evolution
of these complexities by following two different prescriptions viz. Covariant
matrix method and Nielsen's method. Independently, using the two-mode squeezed
states formalism we also compute the R\'enyi and Von-Neumann entanglement
entropy, which show an inherent connection between the entanglement entropy and
quantum circuit complexity. We study the behaviour of the complexity measures
and entanglement entropy separately for three different spatial dimensions and
observe various significant different features in three spatial dimensions on
the evolution of these quantities with respect to the scale factor.
Furthermore, we also study the underlying behaviour of the equilibrium
temperature with two of the most essential quantities i.e. rate of change of
complexity with scale factor and the entanglement entropy. We observe that
irrespective of the spatial dimension, the equilibrium temperature depends
quartically on entanglement entropy.
