Precision frequency metrology and attosecond pulse generation critically rely on stabilization of the carrier-envelope phase (CEP) of mode-locked lasers. So far, only a relatively small class of lasers has been successfully stabilized to warrant phase jitters of a few hundred milliradians as they are required for the generation of an isolated attosecond pulse. For stabilizing certain laser types, the exact reasons for the observed difficulties (or the lack thereof) is only poorly understood. Here we compare the free-running CEP noise of four different lasers, including a femtosecond Ti:sapphire laser and three mode-locked fiber lasers. This study indicates a correlation between amplitude and frequency fluctuations at low Fourier frequencies for essentially all lasers investigated. This finding is well explained with technical noise sources and thermal coupling mechanisms below the upperstate lifetime of the laser gain material. However, for one of the lasers under test, we observe a broadband amplitude-to-phase coupling mechanism well above the upperstate lifetime. This coupling mechanism is related to a dynamical loss modulation. We verify our explanation by numerical simulations, which identify resonances of the saturable absorber mirror as a possible explanation for the coupling mechanism. In case of high modulation depth and resonantly enhanced saturation characteristics, such a saturable absorber can give rise to broadband conversion of spontaneous emission amplitude noise into phase noise, which may cause, in turn, extremely broadband noise signatures, exceeding a megahertz bandwidth.