  The possibility that the asymptotic quasi-normal mode (QNM) frequencies can
be used to obtain the Bekenstein-Hawking entropy for the Schwarzschild black
hole -- commonly referred to as Hod's conjecture -- has received considerable
attention. To test this conjecture, using monodromy technique, attempts have
been made to analytically compute the asymptotic frequencies for a large class
of black hole spacetimes. In an earlier work, two of the current authors
computed the high frequency QNMs for scalar perturbations of $(D+2)$
dimensional spherically symmetric, asymptotically flat, single horizon
spacetimes with generic power-law singularities. In this work, we extend these
results to asymptotically non-flat spacetimes. Unlike the earlier analyses, we
treat asymptotically flat and de Sitter spacetimes in a unified manner, while
the asymptotic anti-de Sitter spacetimes is considered separately. We obtain
master equations for the asymptotic QNM frequency for all the three cases. We
show that for all the three cases, the real part of the asymptotic QNM
frequency -- in general -- is not proportional to ln(3) thus indicating that
the Hod's conjecture may be restrictive.
