© 2019 American Physical Society. We report on nonequilibrium properties of graphene probed by superconducting tunnel spectroscopy. A hexagonal boron nitride (hBN) tunnel barrier in combination with a superconducting Pb contact is used to extract the local energy distribution function of the quasiparticles in graphene samples in different transport regimes. In the cases where the energy distribution function resembles a Fermi-Dirac distribution, the local electron temperature can directly be accessed. This allows us to study the cooling mechanisms of hot electrons in graphene. In the case of long samples (device length L much larger than the electron-phonon scattering length le-ph), cooling through acoustic phonons is dominant. We find a crossover from the dirty limit with a power law T3 at low temperature to the clean limit at higher temperatures with a power law T4 and a deformation potential of 13.3 eV. For shorter samples, where L is smaller than le-ph but larger than the electron-electron scattering length le-e, the well-known cooling through electron out-diffusion is found. Interestingly, we find strong indications of an enhanced Lorenz number in graphene. We also find evidence of a non-Fermi-Dirac distribution function, which is a result of noninteracting quasiparticles in very short samples.