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We have measured the specific heat of PrV$_2$Al$_{20}$ at very lowtemperatures, using high quality single crystals with the residual resistivityratio ~ 20. The high-quality single crystals exhibit clear double transitionsat $T_{\rm Q} =$ 0.75 K and $T^* =$ 0.65 K. These transitions are clearer andshift to higher temperature in higher quality single crystals. Besides, therewas no hysteresis in those transitions in warming and cooling process of theheat capacity measurements. In the ordered state below $T^*$, the specific heatshows a $T^4$ dependence, indicating the gapless mode associated with thequadrupole and/or octupole ordering.

Anomalous specific heat behaviour in the quadrupolar Kondo system PrV2Al20