The process in which a relatively unspecialized cell acquires the specialized features of an astrocyte residing in the forebrain. An astrocyte is the most abundant type of glial cell. Astrocytes provide support for neurons and regulate the environment in which they function. [ GOC:cls GOC:jid GOC:dgh http://purl.obolibrary.org/obo/go/references/0000021 http://www.ncbi.nlm.nih.gov/pubmed/16226447 GOC:dph ]