Egocentric gaze anticipation serves as a key building block for the emerging capability of Augmented Reality. Notably, gaze behavior is driven by both visual cues and audio signals during daily activities. Motivated by this observation, we introduce the first model that leverages both the video and audio modalities for egocentric gaze anticipation. Specifically, we propose a Contrastive Spatial-Temporal Separable (CSTS) fusion approach that adopts two modules to separately capture audio-visual correlations in spatial and temporal dimensions, and applies a contrastive loss on the re-weighted audio-visual features from fusion modules for representation learning. We conduct extensive ablation studies and thorough analysis using two egocentric video datasets: Ego4D and Aria, to validate our model design. We also demonstrate improvements over prior state-of-the-art methods. Moreover, we provide visualizations to show the gaze anticipation results and provide additional insights into audio-visual representation learning.