The material α−RuCl3 has been the subject of intense scrutiny as a potential Kitaev quantum spin liquid, predicted to display Majorana fermions as low-energy excitations. In practice, α−RuCl3 undergoes a transition to a state with antiferromagnetic order below a temperature TN≈7 K, but this order can be suppressed by applying an external in-plane magnetic field of H∥=7 T. Whether a quantum spin liquid phase exists just above that field is still an open question, but the reported observation of a quantized thermal Hall conductivity at H∥>7 T by Kasahara and co-workers [Nature (London) 559, 227 (2018)] has been interpreted as evidence of itinerant Majorana fermions in the Kitaev quantum spin liquid state. In this study, we reexamine the origin of the thermal Hall conductivity κxy in α−RuCl3. Our measurements of κxy(T) on several different crystals yield a temperature dependence very similar to that of the phonon-dominated longitudinal thermal conductivity κxx(T), for which the natural explanation is that κxy is also mostly carried by phonons. Upon cooling, κxx peaks at T≃20 K, then drops until TN, whereupon it suddenly increases again. The abrupt increase below TN is attributed to a sudden reduction in the scattering of phonons by low-energy spin fluctuations as these become partially gapped when the system orders. The fact that κxy also increases suddenly below TN is strong evidence that the thermal Hall effect in α−RuCl3 is also carried predominantly by phonons. This implies that any quantized signal from Majorana edge modes would have to come on top of a sizable—and sample-dependent—phonon background.