The phrase "on/under pain of something" is often used to describe a consequence or penalty that will occur if a certain condition is not met. Some synonyms for this phrase include "subject to," "liable to," "at risk of," "under threat of," and "under the threat of." These phrases all indicate the potential negative outcome that may occur if certain requirements are not fulfilled. They are often used in legal or formal contexts and emphasize the severity of the consequences if the conditions are not met. It is important to note that each of these phrases may have slightly different implications and nuances.