The secular evolution of comet nucleus rotation states subject to outgassing torques is studied. The dynamical model assumes that the nucleus inertia ellipsoid is axially symmetric. The outgassing torques acting on the surface are modeled using standard cometary activity formulae. The general rotational equations of motion are derived and separately averaged over the fast rotational dynamics terms and the comet orbit. Special cases where the averaging assumptions cannot be applied are evaluated separately. The modification of the comet orbit due to comet outgassing is neglected. Resulting from this analysis is a system of secular differential equations that describes the dynamics of the comet nucleus angular momentum and rotation state. We find that the qualitative secular evolution of the rotation state is controlled by a single parameter that combines parameters related to the comet orbit and parameters related to the nucleus surface geometry and activity. From this solution, we find qualitatively different evolutionary paths for comet nuclei whose entire surface is active, as compared to nuclei with only a single active region. For surface activity models between these extremes, we show that certain evolutionary paths are more likely than others. Additionally, our solution indicates that a comet nucleus' rotational angular momentum will tend to increase over time, potentially contributing to the observed phenomenon of comet nucleus splitting.