  In black hole evaporation process, the mass of the hole anti-correlates with
the Hawking temperature. This indicates that the smaller holes have higher
surface gravity. For analogue Hawking effects, however, the acoustic surface
gravity is determined by the local values of the dynamical velocity of the
stationary background fluid flow and the speed of propagation of the
characteristic perturbation embedded in the background fluid, as well as by
their space derivatives evaluated along the direction normal to the acoustic
horizon, respectively. The mass of the analogue system - whether classical or
quantum - does not directly contribute to extremise the value of the associated
acoustic surface gravity. For general relativistic axially symmetric background
fluid flow in the Schwarzschild metric, we show that the initial boundary
conditions describing such accretion influence the maximization scheme of the
acoustic surface gravity and associated analogue temperature. Aforementioned
background flow onto black holes can assume three distinct geometric
configurations. Identical set of initial boundary conditions can lead to
entirely different phase-space behavior of the stationary flow solutions, as
well as the salient features of the associated relativistic acoustic geometry.
This implies that it is imperative to investigate how the measure of the
acoustic surface gravity corresponding to the accreting black holes gets
influenced by the geometric configuration of the inflow described by various
thermodynamic equations of state. Such investigation is useful to study the
effect of Einstenian gravity on the non-conventional classical features as
observed in Hawking like effect in a dispersive medium in the limit of a strong
dispersion relation.
