Higher-order topological phases for time-reversal-symmetry breaking superconductivity in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi mathvariant="normal">UTe</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math>

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 ${B}_{jg}+i{A}_{u}$ ($j,k=1,2,3$; $j\ensuremath{\ne}k$), 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.