As an alternative to state-of-the-art laser frequency stabilization using ultrastable cavities, it has been proposed to exploit the nonlinear effects from coupling of atoms with a narrow transition to an optical cavity. Here, we have constructed such a system and observed nonlinear phase shifts of a narrow optical line by a strong coupling of a sample of strontium-88 atoms to an optical cavity. The sample temperature of a few mK provides a domain where the Doppler energy scale is several orders of magnitude larger than the narrow linewidth of the optical transition. This makes the system sensitive to velocity dependent multiphoton scattering events (Dopplerons) that affect the cavity field transmission and phase. By varying the number of atoms and the intracavity power, we systematically study this nonlinear phase signature which displays roughly the same features as for much lower temperature samples. This demonstration in a relatively simple system opens new possibilities for alternative routes to laser stabilization at the sub-100 mHz level and superradiant laser sources involving narrow-line atoms. The understanding of relevant motional effects obtained here has direct implications for other atomic clocks when used in relation to ultranarrow clock transitions.