In AoA protection mode the sidestick commands an angle of attack. Attainment of the commanded angle of attack will not be instantaneous. A nose-up movement of the sidestick will request a change of the elevator position to pitch the airplane nose-up until the actual AoA is equal to the commanded value. That will take a certain time, depending on the magnitude of the change, the airspeed, the gains in the control loop and the pitch inertia of the airplane. If the airplane is decelerating or accelerating, the phugoid damping comes into play. The accident report on an A320 hard landing at Bilbao in 2001 describes it thus:
AoA protection (high angle of attack protection) ELAC, L80
(...) in these high angle of attack situations, with a tendency toward phugoid movement, studied by longitudinal dynamic stability, in which the aeroplane oscillates between two kinetic and potential energy levels, the EFCS behaves as a damper of the oscillations, commanding appropriate variations of angle of attack in a way that, when the aircraft is slowing down, makes it pitch downward and vice versa.