The evolution of the communication and computation systems enables the user equipment (UE) to handle tremendous transmission data. However, the high-speed data processing also makes UEs release heat and might burn the chips inside the devices. The thermal issue would be more critical in millimeter-wave communications. The massive antenna arrays and the radio frequency modules not only drain the UE battery but also heat the devices. 3GPP also identified the thermal issue and suppressed heat generation by temporarily reducing UE capability in Release 15. In this work, instead of reducing the UE capability, we propose to apply the beam-aware Discontinuous Reception (DRX) mechanism to manage the power consumption and temperature of UEs simultaneously. We are the first to analyze the temperature for UEs with DRX configured. A semi-Markov model is provided, and we employ it to estimate the sleep ratio, packet delay, and steady temperature. We use a simulation program to verify the proposed analytical model. When comparing the beam-aware DRX with the baseline 5G NR DRX operation, we find that the beam-aware scheme reduces the steady temperature from $38.2^\circ C$ to $26.7^\circ C$ . The results show that Beam-Aware DRX could solve the thermal issue without sacrificing much performance of packet delivery latency.