We investigate the nonequilibrium energy transport across a graphene-based unconventional superconductor (SC) hybrid structure with a barrier region sandwiched between normal and superconducting regions, which is evaluated within an extended Dirac Bogoliubov-de Gennes formalism. Three different scenarios of the hybrids having p-wave triplet SCs with (i) p x -, (ii) p y -, and (iii) unitary chiral p x + ip y -parings, are involved. Between them, the thermal conductance, asymmetric Kapitza resistance, and negative differential thermal conductance, respectively, characteristic by much different anomalous thermal properties, are uncovered. More interestingly, thermal energies Klein tunneling also turns up.