High power rechargeable lithium batteries are a key target for transport and load leveling, in order to mitigate CO2 emissions. It has already been demonstrated that mesoporous lithium intercalation compounds (composed of particles containing nanometer diameter pores separated by walls of similar size) can deliver high rate (power) and high stability on cycling. Here we investigate how the critical dimensions of pore size and wall thickness control the rate of intercalation (electrode reaction). By using mesoporous beta-MnO2, the influence of these mesodimensions on lithium intercalation via single and two-phase intercalation processes has been studied in the same material enabling direct comparison. Pore size and wall thickness both influence the rate of single and two-phase intercalation mechanisms, but the latter is more sensitive than the former.