Once one of the most powerful regime figures, Bunyoni was prime minister from mid-2020 until September 2022 when he was fired, days after President Evariste Ndayishimiye had warned of an alleged coup plot against him.
He was sentenced in December to life in prison on a raft of charges including plotting to topple the constitutional regime as well as using witchcraft to threaten the president's life, undermining national security, destabilizing the economy and illegal enrichment.
The army general, now 52, had pleaded not guilty to all counts and said he should be acquitted because of a lack of evidence.
The appeal hearing began amid high security on Monday at the Supreme Court sitting in the prison in the political capital Gitega where he is being incarcerated.
"General Bunyoni once again categorically denied any attempt to overthrow the government," one individual who attended the hearing told AFP on condition of anonymity.
Bunyoni reiterated his position that there was a lack of "material evidence and an absence of legal elements" to support the case against him, according to the witness.
The prosecution has also launched an appeal, asserting that Bunyoni's sentence was too light in terms of fines and the confiscation of assets.
The former prime minister was arrested in April last year in Burundi's main city of Bujumbura.
A former police chief and internal security minister, he was seen as the head of a cabal of military leaders known as "the generals" who wielded the true political power in Burundi.
A close ally of former president Pierre Nkurunziza, Bunyoni was an influential figure in the ruling CNDD-FDD party since it took power in 2005.
Since taking power in June 2020, Ndayishimiye has been lauded by the international community for gradually ending years of Burundi's isolation under Nkurunziza's chaotic and bloody rule.
But he has failed to improve Burundi's dire human rights record and the country remains one of the poorest on the planet.
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