The comb-like spectrum of a white light-illuminated Fabry-P{e}rot etalon can serve as a cost-effective and stable reference for precise Doppler measurements. Understanding the stability of these devices across their broad (100s of nm) spectral bandwidths is essential to realize their full potential as Doppler calibrators. However, published descriptions remain limited to small bandwidths or short timespans. We present a $sim6$ month broadband stability monitoring campaign of the Fabry-P{e}rot etalon system deployed with the near-infrared Habitable Zone Planet Finder spectrograph (HPF). We monitor the wavelengths of each of $sim3500$ resonant modes measured in HPF spectra of this Fabry-P{e}rot etalon (free spectral range = 30 GHz, bandwidth = 820 - 1280 nanometers), leveraging the accuracy and precision of an electro-optic frequency comb reference. These results reveal chromatic structure in the Fabry-P{e}rot mode locations and in their evolution with time. We measure an average drift on the order of 2 cm s $^{-1}$ d$^{-1}$, with local departures up to $pm5$ cm s $^{-1}$ d$^{-1}$. We discuss these behaviors in the context of the Fabry-P{e}rot etalon mirror dispersion and other optical properties of the system, and the implications for the use of similar systems for precise Doppler measurements. Our results show that this system supports the wavelength calibration of HPF at the $lesssim10$ cm s $^{-1}$ level over a night and at the $lesssim30$ cm s $^{-1}$ level over $sim10$ d. Our results also highlight the need for long-term and spectrally-resolved study of similar systems that will be deployed to support Doppler measurement precision approaching $sim10$ cm s $^{-1}$.
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