  We present thermal measurements of large area (over $1,000$~$\mu$m$^2$)
monolayer graphene samples at cryogenic temperatures to study the
electron-phonon thermal conductivity of graphene. By using two large samples
with areas which differ by a factor of 10, we are able to clearly show the area
dependence of the electron-phonon cooling. We find that, at temperatures far
below the Bloch-Gruneisen temperature $T_\mathrm{BG}$, the electron-phonon
cooling power is accurately described by the $T^4$ temperature dependence
predicted for clean samples. Using this model, we are able to extract a value
for the electron-phonon coupling constant as a function of gate voltage, and
the graphene electron-lattice deformation potential. We also present results
for thermal conductance at higher temperatures, above $T_\mathrm{BG}/4$, for
which the clean limit no longer applies. In this regime we find a cooling power
which is accurately described qualitatively, but not quantitatively, by a model
which predicts the emission of very high energy phonons through a
disorder-assisted mechanism.
