We investigate both the longitudinal thermal conductivity (κ_{xx}) and the planar thermal Hall conductivity (κ_{xy}) in the Kitaev spin liquid candidate of the cobalt-based honeycomb antiferromagnet Na_{2}Co_{2}TeO_{6} in a magnetic field (B) applied along the a- and a^{*}-axes. A finite κ_{xy} is resolved for both field directions in the antiferromagnetic (AFM) phase below the Néel temperature of 27 K. The temperature dependence of κ_{xy}/T shows the emergence of topological bosonic excitations. In addition, the field dependence of κ_{xy} shows sign reversals at the critical fields in the AFM phase, suggesting the changes in the Chern number distribution of the topological magnons. Remarkably, a finite κ_{xy} is observed in B∥a^{*} between the first-order transition field in the AFM phase and the saturation field, which is prohibited in a disordered state by the two-fold rotation symmetry around the a^{*} axis of the honeycomb lattice, showing the presence of a magnetically ordered state that breaks the two-fold rotation symmetry. Our results demonstrate the presence of topological magnons in this compound in the whole field range below the saturation field.
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