Temperature evolutions of the electronic specific heat C e (T ) [A. Amon et al., Phys. Rev. B 97, 014501 (2018)] and the thermodynamic critical field B c (T ) [R. Khasanov et al., Phys. Rev. Research 2, 023142 (2020)] of the BeAu superconductor were reconsidered within the framework of the single-gap and two-gap scenarios. The analysis shows that the single-gap approach, by being able to describe satisfactorily the temperature dependence of C e (T ), fails in the case of B c (T ). The self-consistent two-gap model, in contrast, is able to describe both thermodynamic quantities by using a similar set of parameters. Our results imply that in some particular cases, such as in the case of BeAu, measurement of one single thermodynamic quantity may not be enough in order to speculate on the nature of the superconducting pairing mechanism.