  A black hole immersed in a rotating Universe, described by the
Gimon-Hashimoto solution, is tested on stability against scalar field
perturbations. Unlike the previous studies on perturbations of this solution,
which dealt only with the limit of slow Universe rotation j, we managed to
separate variables in the perturbation equation for the general case of
arbitrary rotation. This leads to qualitatively different dynamics of
perturbations, because the exact effective potential does not allow for
Schwarzschild-like asymptotic of the wave function in the form of purely
outgoing waves. The Dirichlet boundary conditions are allowed instead, which
result in a totally different spectrum of asymptotically G\"odel black holes:
the spectrum of quasinormal frequencies is similar to the one of asymptotically
anti-de Sitter black holes. At large and intermediate overtones N, the spectrum
is equidistant in N. In the limit of small black holes, quasinormal modes
(QNMs) approach the normal modes of the empty G\"odel space-time. There is no
evidence of instability in the found frequencies, which supports the idea that
the existence of closed time-like curves (CTCs) and the onset of instability
correlate (if at all) not in a straightforward way.
