An emulator for the Lyman-$\alpha$ forest in beyond-$\Lambda$CDM
  cosmologies
  Interpreting observations of the Lyman-$\alpha$ forest flux power spectrum
requires interpolation between a small number of expensive simulations. We
present a Gaussian process emulator modelling the 1D flux power spectrum as a
function of the amplitude and slope of the small-scale linear matter power
spectrum, and the state of the intergalactic medium at the epoch of interest
($2 < z < 4$). This parameterisation enables the prediction of the flux power
spectrum in extended cosmological models that are not explicitly included in
the training set, eliminating the need to construct bespoke emulators for a
number of extensions to $\Lambda$CDM. Our emulator is appropriate for
cosmologies in which the linear matter power spectrum is described to percent
level accuracy by just an amplitude and slope across the epoch of interest, and
in the regime probed by eBOSS/DESI data. We demonstrate this for massive
neutrino cosmologies, where the emulator is able to predict the flux power
spectrum in a $\Sigma m_\nu=0.3$ eV neutrino cosmology to sub-percent accuracy,
without including massive neutrinos in the training simulations. Further
parameters would be required to describe models with sharp features in the
linear power, such as warm or light axion dark matter. This work will
facilitate the combination of upcoming DESI data with observations of the
cosmic microwave background, to obtain constraints on neutrino mass and other
extensions to $\Lambda$CDM cosmology.
