Chlorosomes are the largest light harvesting complexes in nature and consist of many bacteriochlorophyll pigments forming self-assembled J-aggregates. In this work, we use two-dimensional electronic spectroscopy (2D-ES) to investigate ultrafast dynamics of excitation energy transfer (EET) in chlorosomes and their temperature dependence. From time evolution of the measured 2D electronic spectra of chlorosomes, we directly map out the distribution of the EET rate among the manifold of exciton states in a 2D energy space. In particular, it is found that the EET rate varies gradually depending on the energies of energy-donor and energy-acceptor states. In addition, from comparative 2D-ES measurements at 77 K and room temperature, we show that the EET rate exhibits subtle dependence on both the exciton energy and temperature, demonstrating the effect of thermal excitation on the EET rate. This observation suggests that active thermal excitation at room temperature prevents the excitation trapping at low-energy states and thus promotes efficient exciton diffusion in chlorosomes at ambient temperature.