Costa is the son of two devoted Roman Catholic teachers who lived in Lacló [de], in the then District of Manatuto, where he was born.

Subsequently, after realising that he was unable to fulfil his parents' dream that he would become a priest, he left the seminary and enrolled at the Liceu Dr. Francisco Machado.

[1] By 1978, Costa had become a Fretilin member; that year he witnessed the destruction by the Indonesians of the East Timorese resistance base in Laclo.

"[5] Costa remained in office as Minister of Justice notwithstanding the breakdown of the Alliance for Change and Progress (AMP) coalition during the first few months of 2020, and the consequent restructuring of the government in mid-2020.

[10][11] In addition to his autobiographical book 26 anos, to testemunho (Dili: Livraria Central, 2010, 2nd edition 2013; OCLC 897041092) on the East Timorese struggle for freedom, Costa has written various poems on the same topic.