We demonstrate the emergence of a pronounced thermal transport in the recently discovered class of magnetic materials-altermagnets. From symmetry arguments and first-principles calculations performed for the showcase altermagnet, RuO_{2}, we uncover that crystal Nernst and crystal thermal Hall effects in this material are very large and strongly anisotropic with respect to the Néel vector. We find the large crystal thermal transport to originate from three sources of Berry's curvature in momentum space: the Weyl fermions due to crossings between well-separated bands, the strong spin-flip pseudonodal surfaces, and the weak spin-flip ladder transitions, defined by transitions among very weakly spin-split states of similar dispersion crossing the Fermi surface. Moreover, we reveal that the anomalous thermal and electrical transport coefficients in RuO_{2} are linked by an extended Wiedemann-Franz law in a temperature range much wider than expected for conventional magnets. Our results suggest that altermagnets may assume a leading role in realizing concepts in spin caloritronics not achievable with ferromagnets or antiferromagnets.