  We study a theoretical model for the magnetothermal conductivity of a
spin-1/2 ladder with low exchange coupling ($J\ll\Theta_D$) subject to a strong
magnetic field $B$. Our theory for the thermal transport accounts for the
contribution of spinons coupled to lattice phonon modes in the one-dimensional
lattice. We employ a mapping of the ladder Hamiltonian onto an XXZ spin-chain
in a weaker effective field B_{eff}=B-B_{0}$, where $B_{0}=(B_{c1}+B_{c2})/2$
corresponds to half-filling of the spinon band. This provides a low-energy
theory for the spinon excitations and their coupling to the phonons. The
coupling of acoustic longitudinal phonons to spinons gives rise to
hybridization of spinons and phonons, and provides an enhanced $B$-dependant
scattering of phonons on spinons. Using a memory matrix approach, we show that
the interplay between several scattering mechanisms, namely: umklapp, disorder
and phonon-spinon collisions, dominates the relaxation of heat current. This
yields magnetothermal effects that are qualitatively consistent with the
thermal conductivity measurements in the spin-1/2 ladder compound ${\rm
Br_4(C_5H_{12}N)_2}$ (BPCB).
