Intrusion detection systems (IDS) generate a large number of false alerts which makes it difficult to inspect true positives. Hence, alert prioritization plays a crucial role in deciding which alerts to investigate from an enormous number of alerts that are generated by IDS. Recently, deep reinforcement learning (DRL) based deep deterministic policy gradient (DDPG) off-policy method has shown to achieve better results for alert prioritization as compared to other state-of-the-art methods. However, DDPG is prone to the problem of overfitting. Additionally, it also has a poor exploration capability and hence it is not suitable for problems with a stochastic environment. To address these limitations, we present a soft actor-critic based DRL algorithm for alert prioritization (SAC-AP), an off-policy method, based on the maximum entropy reinforcement learning framework that aims to maximize the expected reward while also maximizing the entropy. Further, the interaction between an adversary and a defender is modeled as a zero-sum game and a double oracle framework is utilized to obtain the approximate mixed strategy Nash equilibrium (MSNE). SAC-AP finds robust alert investigation policies and computes pure strategy best response against opponent's mixed strategy. We present the overall design of SAC-AP and evaluate its performance as compared to other state-of-the art alert prioritization methods. We consider defender's loss, i.e., the defender's inability to investigate the alerts that are triggered due to attacks, as the performance metric. Our results show that SAC-AP achieves up to 30% decrease in defender's loss as compared to the DDPG based alert prioritization method and hence provides better protection against intrusions. Moreover, the benefits are even higher when SAC-AP is compared to other traditional alert prioritization methods including Uniform, GAIN, RIO and Suricata.