The 69-year-old Onehunga man was bashed in the early hours of Saturday December 8. He died several days later after his life support was switched off.
A 29-year-old man is before the courts charged with Brown's murder.
Detective Senior Sergeant Greg Cramer says although his sexuality may have played a role, it is just one of the avenues they are investigating.
He says he does not want to speculate as to motive but is not ruling anything out.