Air molecule is thrown out of the periphery of an impeller at the velocity of the same, which is proportional to its speed in rpm.

Kinetic energy is 1/2mv^2. Square of velocity produces energy. Static pressure = energy.

We have to recognize that static pressure, being a pressure, is a component of energy. It is Force / area. When static pressure is multiplied with airflow, it acquires the dimensions of area x distance. What is produced is Energy. Remember Energy is Force x Distance. The Energy turns into Power as the airflow is volume per unit of time.

As such, static pressure varies at square of fan rpm change.