In this paper, we study the influence of the mortality trade-off in a cane-toad equation, a nonlocal reaction-diffusion-mutation equation modelling the invasion of cane toads in Australia that has attracted attention recently. The population of toads is structured by a phenotypical trait that governs the spatial diffusion. We are concerned with the case when the diffusivity can take unbounded values and the mortality trade-off depends only on the trait variable. Depending on the rate of increase of the penalization from the trade-off as diffusivity approaches infinity, we obtain the rate of spreading of the population. We identify two regimes, an acceleration regime when the penalization is weak and a linear spreading regime when the penalization is strong.