Theoretical models show that in a non-constant environment two predator species feeding on one and the same prey may coexist because the two species occupy different temporal niches: the one with the longer handling time has the advantage when prey is rare so that holding on to the same catch is the better option, while the species with the shorter handling time has the advantage when prey is common and easy to catch. In this paper we address the question whether a predator species with a handling time that is not fixed but decreases with prey density could be selective superior regardless of whether the prey is rare or common, as such predator would be able to occupy both temporal niches all by itself. To that end we study the Rosenzweig-MacArthur model with a modified Holling type II functional response with a density dependent handling time and a handling time dependent conversion factor. We find that the population dynamics tend to be richer than that of the standard model with fixed handling times because of the possibility of multiple positive equilibria and positive attractors. Increasing the strength of the density dependence eventually stabilises the population. Using the framework of adaptive dynamics, we study the evolution of the strength of the density dependence. We find that a predator with even a weakly density dependent handling time can invade both monomorphic and dimorphic populations of predators with fixed handling times. Eventually the strength of the density dependence of the handling time evolves to a level where population cycles are lost, with that the possibility of predator coexistence as well.