Precise measurements of the thermodynamic critical field (B_{c}) in the type-I noncentrosymmetric superconductor BeAu were performed by means of the muon-spin rotation and relaxation technique. The temperature evolution of B_{c} cannot be described within the single-gap scenario and it requires the presence of at least two different types of superconducting order parameters. The self-consistent two-gap approach, adapted for analysis of B_{c}(T) behavior, suggests the presence of two superconducting energy gaps with the gap-to-T_{c} ratios 2Δ/k_{B}T_{c}≃4.52 and ≃2.37 for the big gap and the small gap, respectively Δ=Δ(T=0) is the zero-temperature value of the gap and k_{B} is the Boltzmann constant. This implies that the superconductivity in BeAu is unconventional and that the supercarrier pairing occurs at various energy bands.