While anthropomorphism in human-robot interaction is often discussed, it still appears to lack formal grounds. We recently proposed a first model of the dynamics of anthropomorphism that reflects the evolution of anthropomorphism in the human-robot interaction over time. The model also accounts for non-monotonic effects like the so-called novelty effect. This contribution proposes to build upon this model to investigate the cognitive correlates induced by a sustained human-robot interaction and we present here our initial ideas. We propose to distinguish three cognitive phases: pre-cognitive, familiarity-based, and adapted anthropomorphism, and we outline how these phases relate to the phenomenological evolution of anthropomorphism over time.