1,260. The King and Kohein Gadol Don't Get a Vote

Hilchos Kiddush HaChodesh 4:10

If two judges say that it is necessary to declare a leap year and three disagree, the minority view is discarded. If three say that it is necessary to declare a leap year and two disagree, two more invited judges are added to the court and the matter is discussed. These seven judges should decide; if they all agree to declare a leap year full or not to do so, that decision is followed. If there is a difference of opinion among the judges, we follow the majority, whether the decision is to declare a leap year or not to do so. It is necessary for the head of the Sanhedrin to be one of these seven judges. If the first three judges decide that it is necessary to declare a leap year, it should be done so long as the head of the Sanhedrin is among them or approves. When it comes to declaring a leap year, the opinions of the lower judges are heard first; when it comes to sanctifying a new month, the head of the Sanhedrin is heard first.

Hilchos Kiddush HaChodesh 4:11

Neither a king nor a Kohein Gadol serves on a court to decide whether or not to declare a leap year. A king’s opinion might be influenced by factors affecting his army and their campaigns, while a Kohein Gadol might be concerned about the cold. Specifically, he might be influenced to oppose declaring so that Tishrei will not fall in the winter so that he won’t have to immerse five times on Yom Kippur in the cold.