  In this paper, we investigate analytically the level space of the imaginary
part of quasinormal frequencies for a black hole with a deficit solid angle and
quintessence-like matter by the Padmanabhan's method \cite{Padmanabhan}.
Padmanabhan presented a method to study analytically the imaginary part of
quasinormal frequencies for a class of spherically symmetric spacetimes
including Schwarzschild-de Sitter black holes which has an evenly spaced
structure. The results show that the level space of scalar and gravitational
quasinormal frequencies for this kind of black holes only depend on the surface
gravity of black-hole horizon in the range of -1 < w < -1/3, respectively . We
also extend the range of $w$ to $w \leq -1$, the results of which are similar
to that in -1 < w < -1/3 case. Particularly, a black hole with a deficit solid
angle in accelerating universe will be a Schwarzschild-de Sitter black hole,
fixing $w = -1$ and $\epsilon^2 = 0$. And a black hole with a deficit solid
angle in the accelerating universe will be a Schwarzschild black hole,when
$\rho_0 = 0$ and $\epsilon^2 = 0$. In this paper, $w$ is the parameter of state
equation, $\epsilon^2$ is a parameter relating to a deficit solid angle and
$\rho_0$ is the density of static spherically symmetrical quintessence-like
matter at $r = 1$.
