The three-dimensional (3D) single component multiphase Shan-Chen lattice Boltzmann (LB) model is implemented with the GPU-accelerated algorithm based on the CUDA platform for the simulation of droplet evaporation. It is found that the speed-up of the GPU-accelerated 3D LB model with respect to the CPU-based 3D LB model increases with the computational node number. The maximum speed-up is higher than around 300 when the computational domain is composed of $256^3$ computational nodes. Regarding the calculations performed using the GPU-accelerated 3D LB model that incorporates the operation of data output through CPU, the percentage of computational time consumed by CPU for executing the sequential tasks increases with the computational node number. The model validations are carried out through the comparisons of the simulations with Laplace's law and the $D^2$ law. Then, the GPU-accelerated 3D LB model is applied to simulate the evaporation phenomena of a droplet laying on smooth and rough solid surfaces. It is shown that the contact angle of the evaporating droplet on a smooth solid surface almost remains stable during the evaporation process. For the evaporating droplet sitting on a rough solid surface, however, the time evolution of the contact angle displays obvious oscillations. The simulated morphology evolution of the evaporating droplet sitting on a rough solid surface demonstrates that the oscillations of the contact angle are governed by the stick-slip movement, namely, depinning and receding, of the liquid-gas-solid contact line. This can be attributed to the large hysteresis of the droplet displaying the wetting Wenzel state.