  When measuring the Baryon Acoustic Oscillations (BAO) scale from galaxy
surveys, one typically assumes a fiducial cosmology when converting redshift
measurements into comoving distances and also when defining input parameters
for the reconstruction algorithm. A parameterised template for the model to be
fitted is also created based on a (possibly different) fiducial cosmology. This
model reliance can be considered a form of data compression, and the data is
then analysed allowing that the true answer is different from the fiducial
cosmology assumed. In this study, we evaluate the impact of the fiducial
cosmology assumed in the BAO analysis of the Dark Energy Spectroscopic
Instrument (DESI) survey Data Release 1 (DR1) on the final measurements in DESI
2024 III. We utilise a suite of mock galaxy catalogues with survey realism that
mirrors the DESI DR1 tracers: the bright galaxy sample (BGS), the luminous red
galaxies (LRG), the emission line galaxies (ELG) and the quasars (QSO),
spanning a redshift range from 0.1 to 2.1. We compare the four secondary
AbacusSummit cosmologies against DESI's fiducial cosmology (Planck 2018). The
secondary cosmologies explored include a lower cold dark matter density, a
thawing dark energy universe, a higher number of effective species, and a lower
amplitude of matter clustering. The mocks are processed through the BAO
pipeline by consistently iterating the grid, template, and reconstruction
reference cosmologies. We determine a conservative systematic contribution to
the error of $0.1\%$ for both the isotropic and anisotropic dilation parameters
$\alpha_{\rm iso}$ and $\alpha_{\rm AP}$. We then directly test the impact of
the fiducial cosmology on DESI DR1 data.
