  For any smooth domain $\Omega\subset \mathbb{R}^3$, we establish the
existence of a global weak solution $(\mathbf{u},\mathbf{d}, \theta)$ to the
simplified, non-isothermal Ericksen-Leslie system modeling the hydrodynamic
motion of nematic liquid crystals with variable temperature for any initial and
boundary data $(\mathbf{u}_0, \mathbf{d}_0, \theta_0)\in\mathbf{H}\times
H^1(\Omega, \mathbb{S}^2)\times L^1(\Omega)$, with $
\mathbf{d}_0(\Omega)\subset\mathbb{S}_+^2$ (the upper half sphere) and
$\displaystyle\inf_\Omega \theta_0>0$.
