The phrase "hath lot to do with" implies a strong correlation or connection between two things. There are many synonyms that can be used in place of this phrase, including "is closely linked to," "is heavily influenced by," "has a significant impact on," "is instrumental in," and "plays a key role in." Other options might include "is intertwined with," "is integral to," "is interdependent with," or simply "is connected to." Ultimately, the choice of synonym will depend on the context in which the phrase is being used and the specific nuances of the relationship being described.