We construct quasimodes for the Klein–Gordon equation on the black hole exterior of Kerr–AdS (anti- de Sitter) spacetimes. Such quasimodes are associated with time-periodic approximate solutions of the Klein–Gordon equation and provide natural candidates to probe the decay of solutions on these backgrounds. They are constructed as the solutions of a semiclassical nonlinear eigenvalue problem arising after separation of variables, with the (inverse of the) angular momentum playing the role of the semiclassical parameter. Our construction results in exponentially small errors in the semiclassical parameter. This implies that general solutions to the Klein Gordon equation on Kerr–AdS cannot decay faster than logarithmically. The latter result completes previous work by the authors, where a logarithmic decay rate was established as an upper bound.