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The discovery of unconventional superconductivity in the heavy-fermion material UTe₂ has reinvigorated research of spin-triplet superconductivity. We perform a theoretical study of coupled two-component spin-triplet superconducting order parameters and their thermodynamic transitions into the superconducting state. With focus on the behavior of the temperature dependence of the specific heat capacity, we find that two-component time-reversal symmetry breaking superconducting order may feature vanishing or even negative secondary specific heat anomalies. The origin of this unusual specific heat behavior is tied to the non-unitarity of the composite order parameter. Additionally, we supply an analysis of the topological surface states associated with the different possible spin-triplet orders. The single- and two-component orders carry distinctive anomalous boundary excitations associated to their bulk topology. Depending on the order, we identify second-order topological nodal superconducting phases hosting anomalous surface and hinge states together with the bulk nodes, a Weyl superconducting phase hosting Fermi arcs of Bogoliubov quasiparticles, as well as fully gapped first- and second-order topological phases hosting Majorana Dirac surface states and chiral Majorana hinge modes, respectively.