Is it permissible for a male cousin (son of paternal aunt) to call his female cousin (daughter of paternal aunt) by phone to check on her studies, knowing that this is done with the family's awareness?

Among the objectives of Sharia is to block the means leading to the forbidden (sadd al-dhara'i'). Telephone communication between non-mahram individuals without necessity may lead to what is forbidden. Scholars have explicitly stated the prohibition of a young woman who is feared to cause temptation speaking with a non-mahram man. Therefore, if there is no need for telephone communication between you and your cousin (son of your maternal aunt), it is not permissible for it to continue unless you are both mahram to each other. His desire to check on your studies is not a necessity that warrants such communication, and he can learn about your news from your mother or father.