  Recently an anomalous decay rate of the quasinormal modes of a massive scalar
field in Schwarzschild black holes backgrounds was reported in which the
longest-lived modes are the ones with higher angular number, for a scalar field
mass smaller than a critical value, while that beyond this value the behaviour
is inverted. In this work, we extend the study to other asymptotic geometries,
such as, Schwarzschild-de Sitter and Schwarzschild-AdS black holes. Mainly, we
found that such behaviour and the critical mass are present in the
Schwarzschild-de Sitter background. Also, we found that the value of the
critical mass increases when the cosmological constant increases and also when
the overtone number is increasing. On the other hand, despite the critical mass
is not present in Schwarzschild-AdS black holes backgrounds, the decay rate of
the quasinormal modes always exhibits an anomalous behaviour.
