  Separating out the contributions of different scattering channels in strongly
interacting metals is crucial in identifying the mechanisms that govern their
properties. While momentum or current relaxation rates can be readily probed
via \textit{dc} resistivity or optical/THz spectroscopy, distinguishing
different kinds of inelastic scattering can be more challenging. Using
nonlinear THz 2D coherent spectroscopy, we measure the rates of energy
relaxation after THz excitation in the strongly interacting Fermi liquid,
Sr$_2$RuO$_4$. Energy relaxation is a bound on the total scattering and
specifically a measure of contributions to the electron self-energy that arise
from {\it inelastic} coupling to a bath. We observe two distinct energy
relaxation channels: a fast process that we interpret as energy loss to the
phonon system and a much slower relaxation that we interpret as arising from a
non-equilibrium phonon effects and subsequent heat loss through diffusion.
Interestingly, even the faster energy relaxation rate is at least an order of
magnitude slower than the overall momentum relaxation rate, consistent with
strong electron interactions and the dominance of energy-conserving umklapp or
interband electron-electron scattering in momentum relaxation. The slowest
energy relaxation rate decays on a sub-GHz scale, consistent with the
relaxation dynamics of non-equilibrium phonons. Our observations reveal the
versatility of nonlinear THz spectroscopy to measure the energy relaxation
dynamics in correlated metals. Our work also highlights the need for improved
theoretical understanding of such processes in interacting metals.
