Question: This global-spanning empire is divided among five royal families, each headed by a prince. The emperor himself leads the empire, with the five princes serving under him. While they owe fealty to the king who has absolute authority over the empire, these families operate semi-independently and control their realms directly, as well as their own security forces and economy. The king holds all secular power by law. However, his right to rule is given to him by the gods, with priests of the theocracy speaking on their behalf.

The selection process of becoming king is administered by the priests themselves in order to prove the future king's worthiness. A series of trials are conducted in which the heads of the five families compete against each other. These contests vary with each selection process, and these princes can participate themselves or elect a champion from their realm to compete on their behalf. The winning prince of these contests is then elected to the position of emperor, and rules until he dies. The four losing princes are ritually sacrificed to the gods. This system is meant to prevent any disputes between rival families as to who is the rightful ruler, and to keep the empire stable.

However, it is possible that a prince or several princes may simply not accept the results. They may lead their families into open rebellion to avoid their honorable fate of being sacrificed for the good of the empire. This is unacceptable, for it would tear the empire apart with various wars.

How would I prevent this from happening?

Answer: 
**Prince-in-waiting likes the situation!**

If you are dead set on "prince dies" you make it work by having many princelings in a family. One of these immediately takes over for the family not on the death of the old one, but when the old one loses and is condemned. 

If a given losing prince refuses to accept his demotion from prince to sacrifice, there are people in his court who are eager for him to accept his fate - his heirs / brothers, one of whom is at that moment the new prince, according to the king. Revolution risks everything, and sending an older no-longer-a-prince brother off to die risks little. The military would prefer not to fight 4 other houses and so it is an easy decision for them too. 

This system falls apart if a given prince does not have an heir. Solution - princes from each families line reside safe in the capitol. These princes may or may not be next in line for the throne, but they could be. This is the system of the Romans - raise the children of your enemies as Romans. Then when they go back to rule their people, they will in theory be sympathetic to their overlords.

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**Substitute prince.**

A given prince might try to dodge his fate (if he were pretty certain to lose) by ceding the princedom to some elderly half brother stand-in, drafted into rulership from civilian life and expected in short order to lose and die. The new prince holds his head high and takes it like a man for his country. Unless by chance his champion turns out to be a lot better than anyone expected...
Make it so each prince can command their own army, but the army they command is at an opposite end of the empire from the lands they actually control. This, while confusing, prevents any prince from consolidating military power in their home territory and using it against the emperor.