Chad Kelly, 24, was taken into custody at 1:17 a.m. in Englewood, Colorado.
"I am no longer able to participate in public life," O'Connor wrote in a letter addressed to "friends and fellow Americans."
Cosby was sentenced to three years to 10 years in prison for drugging and sexually assaulting Andrea Constand at the actor's home in suburban Philadelphia in 2004.
Members of the city's gay community consider the flyers — which include an image of an assault rifle similar to the one used in the Pulse shooting — a threat.
Adam Lanza fatally shot 20 first-graders and six educators at Sandy Hook Elementary School in 2012 and killed himself as police arrived.