  We study hot-electron cooling by acoustic and optical phonons in monolayer
MoS$_2$. The cooling power $P$ ($P_e = P/n$) is investigated as a function of
electron temperature $T_e$ (0-500 $\mathrm{K}$) and carrier density $n$
($10^{10}$-$10^{13}$ $\mathrm{cm}^{-2}$) taking into account all relevant
electron-phonon (el-ph) couplings. We find that the cross over from acoustic
phonon dominated cooling at low $T_e$ to optical phonon dominated cooling at
higher $T_e$ takes place at $T_e \sim 50$-$75$ $\mathrm{K}$. The unscreened
deformation potential (DP) coupling to the TA phonon is shown to dominate $P$
due to acoustic phonon scattering over the entire temperature and density range
considered. The cooling power due to screened DP coupling to the LA phonon and
screened piezoelectric (PE) coupling to the TA and LA phonons is orders of
magnitude lower. In the Bloch-Gr\"uneisen (BG) regime, $P\sim T_e^4$ ($T_e^6$)
and $P\sim n^{-1/2}$ ($P_e\sim n^{-3/2}$) are predicted for unscreened
(screened) el-ph interaction. The cooling power due to optical phonons is
dominated by zero-order DP couplings and the Fr\"ohlich interaction, and is
found to be significantly reduced by the hot-phonon effect when the phonon
relaxation time due to phonon-phonon scattering is large compared to the
relaxation time due to el-ph scattering. The $T_e$ and $n$ dependence of the
hot-phonon distribution function is also studied. Our results for monolayer
MoS$_2$ are compared with those in conventional two-dimensional electron gases
(2DEGs) as well as monolayer and bilayer graphene.
