Human adipose-derived stem cells (hASCs) are currently a point of focus for bone tissue engineering applications. However, the ex vivo expansion of stem cells before clinical application remains a challenge. Fetal bovine serum (FBS) is largely used as a medium supplement and exposes the recipient to infections and immunological reactions. In this study, we evaluated the osteogenic differentiation process of hASCs in poly-3-hydroxybutyrate-co-3-hydroxyvalerate (PHB-HV) scaffolds with the osteogenic medium supplemented with pooled allogeneic human serum (aHS). The hASCs grown in the presence of FBS or aHS did not show remarkable differences in morphology or immunophenotype. The PHB-HV scaffolds, which were developed by the freeze-drying technique, showed an adequate porous structure and mechanical performance as observed by micro-computed tomography, scanning electron microscopy (SEM), and compression test. The three-dimensional structure was suitable for allowing cell colonization, which was revealed by SEM micrographs. Moreover, these scaffolds were not toxic to cells as shown by 3-(4,5-dimethylthiazol-2-yl)-2,5-diphenyltetrazolium bromide assay. The differentiation capacity of hASCs seeded on scaffolds was confirmed by the reduction of the proliferation, the alkaline phosphatase (AP) activity, expression of osteogenic gene markers (AP, collagen type I, Runx2, and osteocalcin), and the expression of bone markers, such as osteopontin, osteocalcin, and collagen type I. The osteogenic capacity of hASCs seeded on PHB-HV scaffolds indicates that this scaffold is adequate for cell growth and differentiation and that aHS is a promising supplement for the in vitro expansion of hASCs. In conclusion, this strategy seems to be useful and safe for application in bone tissue engineering.