The thermal Hall effect (THE) is a powerful probe of material properties, even in insulators. Here, we investigate the thermal response of altermagnets by developing a theory on Berry curvature driven nonlinear THE from both electrons and magnons. We identify symmetry conditions under which the third-order THE dominates while linear and second-order contributions vanish. Notably, $d$-wave and $i$-wave altermagnets with out-of-plane Néel order satisfy these conditions, whereas in-plane order yields a dominant linear THE. Using ${\mathrm{KV}}_2{\mathrm{Se}}_2\mathrm{O}$ (a $d$-wave altermagnetic metal) and ${\mathrm{MnF}}_2$ (an altermagnetic insulator) as concrete examples, we show nonzero third-order nonlinear THE from electrons and magnons, respectively, both exhibiting a $\pi$-periodic dependence on the direction of temperature gradient. These symmetry-guaranteed Berry curvature phenomena provide a diagnostic for altermagnetism in candidate materials and enable the determination of the Néel vector orientation.

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• Altermagnetism
• Thermal Hall effect
• Altermagnets
• Symmetries
• Boltzmann theory
• First-principles calculations
• Holstein-Primakoff method
• Spin wave theory