The recent discovery of heavy-fermion superconductor ${\mathrm{UTe}}_2$ has broadened the possibility of realizing exotic time-reversal-symmetry breaking superconductivity. However, a comprehensive understanding of the topological phases in the superconducting states of ${\mathrm{UTe}}_2$ is still lacking. Here, we present an exhaustive classification of topological phases for all time-reversal-symmetry breaking pairing symmetries of ${\mathrm{UTe}}_2$. Using the K-theoretical classification approach, we uncover that 25 of 36 possible pairing states are classified as higher-order topological phases, with some demonstrating hybrid-order topology through an intricate interplay of hinge and corner states. Furthermore, under the weak-coupling condition of the pair potentials, the possible pairing symmetries are constrained to $B_{ju}+i{B}_{ku}, A_u+i{B}_{ju}$, and $B_{jg}+i{A}_u$ ($j,k=1,2,3$; $j\ne k$), where these symbols denote the irreducible representations of the point group $D_{2h}$. For these pairing states, the topological invariants are related to the Fermi surface topology via the Fermi-surface formula, enabling us to systematically diagnose higher-order topological phases. Using a tight-binding model, we demonstrate the higher-order topological phases of the mixed-parity $A_u+i{B}_{1g}$ and nonunitary-odd-parity $A_u+i{B}_{1u}$ superconductors, where the second-order and hybrid-order topological phases emerge as the number of Fermi surfaces enclosing the time-reversal invariant momentum evolves from two to four. The findings suggest that ${\mathrm{UTe}}_2$ serves as a compelling platform for exploring higher-order topological superconductors with diverse topological surface states.

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• Majorana fermions
• Topological phases of matter
• Topological superconductors
• Heavy-fermion systems