  We investigate the performance of microwave-frequency phononic crystal
resonators fabricated on thin-film lithium niobate for integration with
superconducting quantum circuits. For different design geometries at
millikelvin temperatures, we achieve mechanical internal quality factors $Q_i$
above $10^5 - 10^6$ at high microwave drive power, corresponding to
$5\times10^6$ phonons inside the resonator. By sweeping the defect size of
resonators with identical mirror cell designs, we are able to indirectly
observe signatures of the complete phononic bandgap via the resonators'
internal quality factors. Examination of quality factors' temperature
dependence shows how superconducting and two-level system (TLS) loss channels
impact device performance. Finally, we observe an anomalous low-temperature
frequency shift consistent with resonant TLS decay and find that material
choice can help to mitigate these losses.
