We develop a theoretical formalism to model the linear spectrum of a quantum dot embedded in a high-quality cavity, in the presence of an arbitrary mechanism modifying the homogeneous spectrum of the quantum dot. Within the simple assumption of Lorentzian broadening, we show how the known predictions of cavity quantum electrodynamics are recovered. We then apply our model to the case where the quantum dot interacts with an acoustic-phonon reservoir, producing phonon sidebands in the response of the bare dot. In this case, we show that the sidebands can sustain the spectral response of the cavitylike peak even at moderate dot-cavity detuning, thus, supporting recent experimental findings.