Mr Pyne told ABC's Q&A on Monday night he would "like to think" he was still friends with former Prime Minister Tony Abbott, who had accused him of being "disloyal".
The recording, reported by News Corp columnist Andrew Bolt, captured Mr Pyne celebrating the success of the left faction of the Liberal Party during its federal council meeting on Friday night.
On Monday, Mr Abbott reminded 2GB's Ray Hadley that Mr Pyne was not just in cabinet, but also a member of the leadership team when he was in office.
"It's important that you show loyalty," he said.
"But if (Mr Pyne is) to be believed on Friday night that loyalty was never there which is incredibly disappointing."
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Responding to Abbot's claim, Mr Pyne told Q&A he had worked hard to get Mr Abbott into the prime ministership.
"I, along with Peta Credlin, Andrew Hirst, Mark Textor, Julie Bishop (and) a couple of other people, were absolutely central to him becoming prime minister and defeating the Labor Party in 2013, and almost defeating them in 2010," he said.
"So nobody could ever question my determination to see a coalition government in power."
"Of course, leaders change in political parties and my vote for Malcolm Turnbull over my whole career is not a secret.
"But obviously when Tony Abbott was the leader, I got up every morning, usually at 6am, to do telephone conference calls everyday, to work out how we could get rid of the Rudd-Gillard government and we did.
"That was for Tony Abbott, but more than Tony Abbott, it was for the Australian people, because I believed we were getting rid of a bad government and putting in a better government, with Tony Abbott as prime minister."
Mr Pyne also addressed his comments in the leaked tapes that marriage equality might "be sooner than everyone thinks".
That led to speculation the government has imminent plans to legalise , with Mr Abbott saying any move to dump the plebiscite would be a breach of faith with the people.
Mr Pyne told Q&A there was no change of policy planned.
"The only thing that's made that statement remarkable is that there's apparently a secret leaked tape, which has excited the press and has turned it into a sensational story," he said.
"But actually, obviously I support the policy. I want there to be a vote of the people so that they are part of that outcome. That was a policy we put to the parliament. Labor voted it down. If they'd voted for the plebiscite, we could have had marriage equality in February or March of 2017."
