The influence of sliding speed in the nanoscale friction forces between a silicon tip and monolayer and multilayer graphene were investigated with the use of an atomic force microscope. We found that the friction forces increase linearly with the logarithm of the sliding speed in a highly layer-dependent way. The increase in friction forces with velocity is amplified at the monolayer. The amplification of the friction forces with velocity results from the introduction of additional corrugation in the interaction potential driven by the tip movement. This effect can be interpreted as a manifestation of local thermally induced surface corrugations in nanoscale influencing the hopping dynamics of the atoms at the contact. These experimental observations were explained by modeling the friction forces with the thermally activated Prandtl-Tomlinson model. The model allowed determination of the interaction potential between tip and graphene, critical forces, and attempt frequencies of slip events. The latter was observed to be dominated by the effective contact stiffness and independent of the number of layers.