According to the Entropy Rate Constancy (ERC) principle, the information density of a text is approximately constant over its length. Whether this principle also applies to nonverbal communication signals is still under investigation. We perform empirical analyses of video-recorded dialogue data and investigate whether listener gaze, as an important nonverbal communication signal, adheres to the ERC principle. Results show (1) that the ERC principle holds for listener gaze; and (2) that the two linguistic factors syntactic complexity and turn transition potential are weakly correlated with local entropy of listener gaze.