The Planum Temporale (PT) is one of the key hubs of the language network in the human brain. The gross asymmetry of this perisylvian region toward the left brain was considered as the most emblematic marker of hemispheric specialization of language processes in the brain. Interestingly, this neuroanatomical signature was documented also in newborn infants and preterms, suggesting the early brain's readiness for language acquisition. Nevertheless, this latter interpretation was questioned by a recent report in nonhuman primates of a potential similar signature in newborn baboons Papio anubis based on PT surface measures. Whether this "tip of the iceberg" PT asymmetry is actually reflecting asymmetry of its underlying grey matter volume remain unclear but critical to investigate potential continuities of cortical specialization with human infants. Here we report a population-level leftward asymmetry of the Planum Temporale grey matter volume in in vivo 34 newborn baboons Papio anubis, which showed intra-individual positive correlation with PT surface's asymmetry measures but also a more pronounced degree of leftward asymmetry at the population-level. This finding demonstrates that PT leftward structural asymmetry in this Old World monkey species is a robust phenomenon in early primate development, which clearly speaks for a continuity with early human brain specialization. Results also strengthen the hypothesis that early PT asymmetry might be not a human-specific marker for the pre-wired language-ready brain in infants.