  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
(100's 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 $^{-1}$.
