Excommunicate. At least when he is using religion to strengthen his reign. Many tyrants have professed to do what they do in the name of a god. That venue can be cut of by the religious leaders.
That was precisely the method utilized by St. Ambrose, bishop of Milan, in 390 CE when the Roman Emperor Theodosius orchestrated a massacre against the inhabitants of Thessalonica who had risen in revolt against the imperial throne.
The entire uprising had been provoked by an incident involving the arrest and detainment of a 'charioteer' popular with the common people, on charges of having committed a "
homosexual offence" (I'm afraid to say he was incarcerated for that). The populace demanded his release. Upon the refusal of the military governor of the province, a general insurrection broke out and in retaliatian the Emperor Theodosius dispatched multiple army units which captured the rebellious city and systematically slaughtered 7,000 of its citizens without trial in a mass extra-judicial war crime.
St. Ambrose, then the Emperor's closest confidante in the church's clerical hierarchy, was appalled by the savagery and unlawful nature of the imperially-sanctioned massacre, to such an extent that he
excommunicated the Emperor:
Massacre of Thessalonica - Wikipedia
Ambrose, the bishop of Milan, after hearing about the massacre, left Milan (which was the residence of Theodosius at that time) and refused to celebrate a Mass in the Emperor's presence, until Theodosius repented. In a letter to the emperor, Ambrose explained his position and gave reasons for his resolution:
What could I do? Should I not hear? But I could not clog my ears with wax, as old fables tell. Should I then speak about what I heard? But I was obliged to avoid precisely what I feared could be brought about by your orders, that is, a bloodshed. Should I remain silent? But then the worst thing would happen as my conscience would be bound and my words taken away. And where would they be then? When a priest does not talk to a sinner, then the sinner will die in his sin, and the priest will be guilty because he failed to correct him.[3]
According to Theodoret, when the emperor tried to enter a Milanese church, where Ambrose was about to celebrate a Mass, the bishop stopped him and rebuked him for what he had done. And because the emperor “had been brought up according to divine words and understood well that some affairs are handled by priests, others by emperors”, he could do nothing but return "weeping and sighing" to the palace.[4]
Eight months had passed and Theodosius still sat in the palace, moaning and sobbing. His magister officiorum Rufinus, who "used great freedom of speech due to the familiarity with the emperor", noticed this behaviour, approached and asked him why he was weeping. Having been told, he volunteered to see the bishop and ask him to reconsider. Theodosius hesitantly agreed and even chose to follow Rufinus from a distance.
Ambrose was not restrained at all when negotiating with Rufinus, scolding him and even accusing him of complicity in the massacre: "Rufinus, you are as impudent as a dog, because it was you who advised the emperor such a bloodshed." When the emperor showed up, Ambrose at first remained stubborn and changed his mind only after Theodosius promised to promulgate a law, which in cases of death sentences would introduce a thirty-day lag before the execution.[5]
And so, as a result of St. Ambrose's passive 'resistance' through the supreme instrument of ecclesiastical excommunication, the Emperor was eventually reduced to a sobbing, penitent 'man-child' and agreed to change the law (to make it that no one could be executed without trial and a thirty-day period in which to make amends for alleged offences) in return for being re-admitted to church.
When he was again permitted to attend Mass, Ambrose compelled him to get down on his knees in a public demonstration of his desire for forgiveness and then sit at the 'back' of the pews with the rest of his lay subjects like a regular joe. The purpose was to show to all and sundry that even the Emperor was subject to the same 'rule of law' in divine eyes, being no greater in worth under God's estimation than any other layperson:
"...The Emperor listened to this advice, and deeming it excellent, he at once ordered the law to be drawn up, and himself signed the document.
St. Ambrose then unloosed his bonds. The Emperor, who was full of faith, now took courage to enter holy church, [where] he prayed neither in a standing, nor in a kneeling posture, but throwing himself on the ground. He tore his hair, struck his forehead, and shed torrents of tears, as he implored forgiveness of God. [Ambrose restored him to favor, but forbade him to come inside the altar rail, ordering his deacon to say], "The priests alone, O Emperor, are permitted to enter within the barriers by the altar. Retire then, and remain with the rest of the laity. A purple robe makes Emperors, but not priests." . . .
[Theodosius uttered some excuses, and meekly obeyed, praising Ambrose for his spirit, and saying], "Ambrose alone deserves the title of 'bishop.'"..."
(William Stearns Davis, Readings in Ancient History, vol II: Rome and the West (Boston: Allyn and Bacon, 1913), pp. 298-300.)
If the emperor in question 'believes' in the religious creed (or has to feign such for reasons of maintaining his rule over Christian subjects in a society founded upon religious belief),
that's how its done. Get the guy on all-fours in sackcloth and ashes, and compel him to institute legal change so that the unlawful, tyrannous acts are safeguarded against in future.
Should he refuse to comply, the excommunication automatically relieves his Christian subjects of any oaths of fealty to him under canon law, meaning that they can all legitimately rise in organised rebellion against his rule and put someone else on the throne in his stead.