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We present experiments on the magnetic-field-dependent thermal transport in the spin-1/2 ladder system (C5H12N)(2)CuBr4. The thermal conductivity kappa(B) is only weakly affected by the field-induced transitions between the gapless Luttinger-liquid state realized for B-c1 < B < B-c2 and the gapped states, suggesting the absence of a direct contribution of the spin excitations to the heat transport. We observe, however, that the thermal conductivity is strongly suppressed by the magnetic field deeply within the Luttinger-liquid state. These surprising observations are discussed in terms of localization of spinons within finite ladder segments and spinon-phonon umklapp scattering of the predominantly phononic heat transport.