The ratio of baryonic to dark matter densities is assumed to have remained constant throughout the formation of structure. With this, simulations show that the fraction f gas ( z ) of baryonic mass to total mass in galaxy clusters should be nearly constant with redshift z . However, the measurement of these quantities depends on the angular distance to the source, which evolves with z according to the assumed background cosmology. An accurate determination of f gas ( z ) for a large sample of hot ( kT e >5 keV), dynamically relaxed clusters could therefore be used as a probe of the cosmological expansion up to z <2. The fraction f gas ( z ) would remain constant only when the correct cosmology is used to fit the data. In this paper, we compare the predicted gas mass fractions for both Λ cold dark matter ( Λ CDM) and the R h = ct Universe and test them against the three largest cluster samples (LaRoque et al. 2006 Astrophys. J. 652, 917–936 ( doi:10.1086/508139 ); Allen et al. 2008 Mon. Not. R. Astron. Soc. 383, 879–896 ( doi:10.1111/j.1365-2966.2007.12610.x ); Ettori et al. 2009 Astron. Astrophys. 501, 61–73 ( doi:10.1051/0004-6361/200810878 )). We show that R h = ct is consistent with a constant f gas in the redshift range z ≲ 2 , as was previously shown for the reference Λ CDM model (with parameter values H 0 =70 km s −1 Mpc −1 , Ω m =0.3 and w Λ =−1). Unlike Λ CDM, however, the R h = ct Universe has no free parameters to optimize in fitting the data. Model selection tools, such as the Akaike information criterion and the Bayes information criterion (BIC), therefore tend to favour R h = ct over Λ CDM. For example, the BIC favours R h = ct with a likelihood of approximately 95% versus approximately 5% for Λ CDM.