The relationship between brain entropy (BEN) and early brain development has been established through animal studies. However, it remains unclear whether the BEN can be used to identify age-dependent functional changes in human neonatal brains and the genetic underpinning of the new neuroimaging marker remains to be elucidated. In this study, we analyzed resting-state fMRI data from the Developing Human Connectome Project, including 280 infants who were scanned at 37.5-43.5 weeks postmenstrual age. The BEN maps were calculated for each subject, and a voxel-wise analysis was conducted using a general linear model to examine the effects of age, sex, and preterm birth on BEN. Additionally, we evaluated the correlation between regional BEN and gene expression levels. Our results demonstrated that the BEN in the sensorimotor-auditory and association cortices, along the 'S-A' axis, was significantly positively correlated with postnatal age (PNA), and negatively correlated with gestational age (GA), respectively. Meanwhile, the BEN in the right rolandic operculum correlated significantly with both GA and PNA. Preterm-born infants exhibited increased BEN values in widespread cortical areas, particularly in the visual-motor cortex, when compared to term-born infants. Moreover, we identified five BEN-related genes (DNAJC12, FIG4, STX12, CETN2, and IRF2BP2), which were involved in protein folding, synaptic vesicle transportation and cell division. These findings suggest that the fMRI-based BEN can serve as an indicator of age-dependent brain functional development in human neonates, which may be influenced by specific genes.