  Stars with accurate and precise effective temperature (T$_{\rm eff}$)
measurements are needed to test stellar atmosphere models and calibrate
empirical methods to determine T$_{\rm eff}$. There are few standard stars
currently available to calibrate temperature indicators for dwarf stars. Gaia
parallaxes now make it possible, in principle, to measure T$_{\rm eff}$ for
many dwarf stars in eclipsing binaries. We aim to develop a method that uses
high-precision measurements of detached eclipsing binary stars, Gaia parallaxes
and multi-wavelength photometry to obtain accurate and precise fundamental
effective temperatures that can be used to establish a set of benchmark stars.
We select the well-studied binary AI Phoenicis to test our method, since it has
very precise absolute parameters and extensive archival photometry. The method
uses the stellar radii and parallax for stars in eclipsing binaries. We use a
Bayesian approach to obtain the integrated bolometric fluxes for the two stars
from observed magnitudes, colours and flux ratios. The fundamental effective
temperature of two stars in AI Phoenicis are $6199\pm22$ K for the F7V
component and $5094\pm16$ K for the K0IV component. The zero-point error in the
flux scale leads to a systematic error of only 0.2% ($\approx$ 11K) in T$_{\rm
eff}$. We find that these results are robust against the details of the
analysis, such as the choice of model spectra. Our method can be applied to
eclipsing binary stars with radius, parallax and photometric measurements
across a range of wavelengths. Stars with fundamental effective temperatures
determined with this method can be used as benchmarks in future surveys.
