The phrase "pivot on" can be replaced by different words depending on the context. Some synonyms for pivot on are depend on, revolve around, center on, rest on, and focus on. These alternatives can be used in different situations depending on the subject matter. For example, if you are discussing a business strategy, you can say that the success of the plan depends on the implementation process. If you are writing a story, you can say that the plot revolves around the protagonist's journey. Additionally, if you are discussing a scientific theory, you can say that the hypothesis centers on a specific idea or concept. These synonyms can help you to add variety and complexity to your writing or conversation.