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Most centrifugal pendulum vibration absorber (CPVA) research focuses on the horizontal or vertical plane, ignoring the influence of gravity. However, with the wide application of CPVAs in the automobile industry, some gravity-related problems have been encountered in practice. In this study, employing the second kind of Lagrange equation, the differential equation of motion of a CPVA is established, and the first-order approximate analytical solution is solved using the method of multiple scales. The mathematical relations among the excitation torque amplitude and phase, gravity influence, absorber trajectory shape, absorber position, viscous damping coefficient, and mistuning level parameters are provided for study. Specifically, the second-order responses of four absorbers and two absorbers in a gravity field are studied, and the influence of the change in the torque excitation phase on the response of the absorber is thoroughly analyzed.