The energy band theory of acoustic crystal provides an important theoretical foundation for controlling the features of sound fields. By utilizing the acoustic flat bands, the sound wave can be effectively modulated to realize the acoustic localization and diffusion. In this work, an artificial gauge field is used to design a system supporting multiple acoustic flat bands, leading to the emergence of diversified acoustic localizations. Initially, cavity resonators, linked with different connectivity based on the field profiles of acoustic resonators, are employed to emulate coupled pz-dipole modes of atomic orbitals. According to the band order of in-phase and out-of-phase modes in two coupled cavities, it can be confirmed that the cross-linked and V-shaped-linked tube structures can achieve the positive coupling and negative coupling, respectively. By introducing positive and negative coupling into a rhombic loop, a synthetic gauge field can be formed due to the $ \pi $ flux phase accumulation of acoustic wave in the loop. Correspondingly, the different geometric phases of acoustic wave in different paths are analogous to the Aharonov-Bohm caging effect. Due to the Aharonov-Bohm caging effect, the introduce of $ \pi $-flux into a rhombic loop causes the dispersion bands to collapse into dispersionless flat bands, providing the opportunity for controlling the localizations of sound fields. According to the finite structures of the cases with and without gauge fluxes, the eigenmodes and energy ratios are analyzed to investigate the sound field distributions. Compared with the zero-flux structure, the acoustic localization can be realized in the bulk and edge of a finite rhombic sonic crystal after introducing an artificial gauge field with a $ \pi $ flux in each plaquette. Here the localized states, induced by Aharonov-Bohm caging effect, are topologically immune to symmetrical structure disorder, indicating that the localized mode relies on the topological feature of the $ \pi $-flux artificial gauge field. Additionally, based on the excitation of flat band eigenstates, the acoustic flat band bound states corresponding to different eigenstates can be obtained. By superimposing acoustic flat band bound states, the amplitude and phase of sound wave can be manipulated at specific locations, realizing the composite flat band bound states with rich acoustic field patterns. Therefore, we achieve different types of acoustic localized states in an acoustic topological Aharonov-Bohm cage. These localized states can be excited in any primitive cell of the rhombic lattices, and possess the remarkable ability to trap sound waves at different bulk gap frequencies, which achieves the broadband sound localizations. At the eigenfrequencies of flat bands, the localized states will be transformed into the extended states, exhibiting acoustic filtering functionality. Therefore, the acoustic Aharonov-Bohm cage is promising for applications at bandgap and flat band frequencies. The findings in this work provide the theoretical guidance for exploring the acoustic localized states with artificial gauge field, and can realize potential applications in acoustic control devices.
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