  Monolayer molybdenum trioxide (MoO$_3$) is an emerging two-dimensional (2D)
material with high electrical conductivity. Using first-principles calculations
and a Boltzmann transport theoretical framework, we predict record low
room-temperature phonon thermal conductivity ($\kappa_p$) of 1.57 W/mK and 1.26
W/mK along the principal in-plane directions of MoO$_3$ monolayer. The behavior
is attributed to the combination of soft flexural and in-plane acoustic modes,
which are coupled through the finite layer thickness, and to the strong bonding
anharmonicity, which gives rise to significant 3- and 4-phonon scattering
events. These insights suggest new indicators for guiding the search of 2D
materials with low $\kappa_p$. Our result motivates experimental $\kappa_p$
measurements in MoO$_3$, and its applications as a thermoelectric and thermally
protective material.
