The square-root operation provides a new scheme to create topological phases with unconventional spectrum properties. With the square-root operation, the square-root topological insulators can support paired topological boundary states in two bulk gaps, and the mechanism of square-root has been generalized to 2n-root topological insulators. In this study, we describe the acoustic realization of third-order quartic-root topological insulators based on the original three-dimensional (3D) square-root sonic crystals. By inserting extra sites into the 3D square-root lattice, we can renormalize the coupling parameters and obtain multiple topological boundary states in different bulk gaps with distinct phase profiles. The topological origin is clearly elucidated with the direct sum relation for the 3D quartic-root lattice. We further validate the robustness of the corner states under random bulk disorder and show the diversified localizations of topological edge states at distinct frequencies on different-shaped 3D sonic crystals. Our work extends the quartic-root topological states into a 3D acoustic system and may find potential applications in multi-frequency acoustic devices.
