We study the Berry curvature effects of the Bogoliubov quasiparticles in chiral superconducting rhombohedral graphene. Using a two-band Bogoliubov-de Gennes Hamiltonian to describe the superconducting quasiparticles, we calculate the momentum-space Berry curvature, orbital magnetic moment, and anomalous thermal Hall, spin Nernst, and orbital Nernst transport of chiral p-wave superconducting states. We investigate the impact of normal-state band warping in rhombohedral graphene, which can generate Bogoliubov Fermi surfaces for quasiparticle excitations. We find that Bogoliubov Fermi surfaces qualitatively modify the anomalous transport responses, inducing a deviation of the thermal Hall conductivity from the quantized value and strongly enhancing the spin and orbital Nernst responses.