Paramedics were called after 10pm and a man was pronounced dead at the venue, a spokeswoman for South Central Ambulance Service said.
Cognito, 60, reportedly “sat on a chair and laid back for five minutes” during the show.
“Everyone in the crowd,
“Even when I walked on stage and touched his arm I was expecting him to say ‘boo’.”
Back in 1984, comedian Tommy Cooper died after collapsing during a performance on live television. On that occasion too, people reported believing the incident was part of his act.
Katy Brand and Rufus Hound have led tributes to Cognito, whose real name was Paul Barbieri.
Actor and comedian Hound said comedy had lost “one of the greats”.
Brand paid tribute to “Cogs”, as he was known on the comedy circuit, saying: “Sad to hear about Ian Cognito.
“I hung with him a good few times in the past, and he was always fascinating and hilarious company. RIP Cogs.”
Tiffany Stevenson, who has appeared on Russell Howard’s Good News and Mock The Week, celebrated Cognito’s reputation as a hell-raiser.
She said: “There has never been more apocryphal tales around a comedian than Ian Cognito.
“Wondering/hoping that at the wake someone will bang a nail into the wall and hang a coat on it.”
Stand-up Luisa Omielan said: “Just heard about Ian Cognito, I saw him once in stage years ago and was in awe, he was as epic as his reputation. Rebellious and brilliant.
“I heard he passed away after being on stage. He even died like a legend. Far far too soon. Love to his family.”
Cognito enjoyed cult status among his contemporaries and won the Time OutAward for Stand-up Comedy in 1999.