  This work makes the first ever attempt to understand the influence of the
black hole background space-time in determining the fundamental properties of
the embedded relativistic acoustic geometry. To accomplish such task, the role
of the spin angular momentum of the astrophysical black hole (the Kerr
parameter $a$ -- a representative feature of the background black hole metric)
in estimating the value of the acoustic surface gravity (the representative
feature of the corresponding analogue space time) has been investigated for
axially symmetric inflow of hydrodynamic fluid onto a rotating black hole.
Since almost all astrophysical black holes are supposed to posses some degree
of intrinsic rotation, the influence of the Kerr parameter on classical
analogue models is very important to understand.
  For certain values of the initial boundary conditions describing the
aforementioned flow, more than one acoustic horizons, namely two black hole
type and one white hole type, may form, where the surface gravity may become
formally infinite at the acoustic white hole. The connection between the
corresponding analogue Hawking temperature with astrophysically relevant
observables associated with the spectral signature has been discussed.
