Quantum acoustodynamics (QAD) is a rapidly developing field of research, offering possibilities to realize and study macroscopic quantum-mechanical systems in a new range of frequencies, and implement transducers and new types of memories for hybrid quantum devices. Here we propose a novel design for a versatile diamond QAD cavity operating at GHz frequencies, exhibiting effective mode volumes of about $10^{-4}\lambda^3$... Our phononic crystal waveguide cavity implements a non-resonant analogue of the optical lightning-rod effect to localize the energy of an acoustic mode into a deeply-subwavelength volume. We demonstrate that this confinement can readily enhance the orbit-strain interaction with embedded nitrogen-vacancy (NV) centres towards the high-cooperativity regime, and enable efficient resonant cooling of the acoustic vibrations towards the ground state using a single NV. This architecture can be readily translated towards setup with multiple cavities in one- or two-dimensional phononic crystals, and the underlying non-resonant localization mechanism will pave the way to further enhance optoacoustic coupling in phoxonic crystal cavities. (read more)