  It is presented a thorough analysis of scalar perturbations in the background
of Gauss-Bonnet, Gauss-Bonnet-de Sitter and Gauss-Bonnet-anti-de Sitter black
hole spacetimes. The perturbations are considered both in frequency and time
domain. The dependence of the scalar field evolution on the values of the
cosmological constant $\Lambda$ and the Gauss-Bonnet coupling $\alpha$ is
investigated. For Gauss-Bonnet and Gauss-Bonnet-de Sitter black holes, at
asymptotically late times either power-law or exponential tails dominate, while
for Gauss-Bonnet-anti-de Sitter black hole, the quasinormal modes govern the
scalar field decay at all times. The power-law tails at asymptotically late
times for odd-dimensional Gauss-Bonnet black holes does not depend on $\alpha$,
even though the black hole metric contains $\alpha$ as a new parameter. The
corrections to quasinormal spectrum due to Gauss-Bonnet coupling is not small
and should not be neglected. For the limit of near extremal value of the
(positive) cosmological constant and pure de Sitter and anti-de Sitter modes in
Gauss-Bonnet gravity we have found analytical expressions.
