The Church did not burn Giordano Bruno at the stake, the State did. The Church tried to save his life. In fact, according to the records of the trial against Giordano Bruno, his sentencing included these words:
“ti scacciamo, dal foro nostro ecclesiastico et dalla nostra santa et immacolata Chiesa, della cui misericordia ti sei reso indegno; et dover esser rilasciato alla Corte secolare, sì come ti rilasciamo alla Corte di voi monsignor (name removed by moderator)e di Roma qui presente, per punirti delle debite pene, pregandolo però efficacemente che voglia mitigare il rigore delle leggi circa la pena della tua persona, che sia senza pericolo di morte o mutilatione di membro.” (The Sentence of Giordano Bruno as delivered to Governor Ferrante Taverna by Flamino Adriano the Notary of the Inquisition of Venice)
That is: “We cast you out from our ecclesiastical forum and from our holy and immaculate Church, of whose mercy you have been made unworthy; and since we must deliver you to the secular court, we release you to the Court of the good gentleman the Governor of Rome here present, to be punished with heavy penalties, but efficaciously pleading that he may mitigate the rigor of the laws concerning the punishment of your person, that it may be without danger of death or mutilation of your members.” (This translation is my own and is likely imperfect. Corrections appreciated.)
As with all trials by Inquisition, Church authorities were forbidden by Canon Law either from executing anyone or from requiring a sentence of death. (Corpus Juris Canonici Book 3 Title 50 Chapter 9) Instead, they were required to release convicted heretics to the secular court and to make an effort to save the lives of convicted heretics and to plead against their execution. (Directory of the Inquisition Section 3a by Inquisitor General Nicholas Eymeric) As recorded in the proceedings of the trial of Giordano Bruno, they followed this requirement and, as quoted above, pleaded for his life in the presence of the governor asking him not to execute him. The Governor of Rome – by his civil authority, not by Church authority – executed him four days later.