In strongly correlated systems, interactions give rise to critical fluctuations surrounding the quantum critical point (QCP) of a quantum phase transition. Quasicrystals allow the study of quantum critical phenomena in aperiodic systems with frustrated magnetic interactions. Here, we study the magnetic field and temperature scaling of the low-temperature specific heat for the quantum critical Yb-Au-Al quasicrystal. We devise a scaling function that encapsulates the limiting behaviors as well as the area where the system goes from a temperature-limited to a field-limited quantum critical region, where the magnetic field acts as a cutoff for critical fluctuations. The zero-field electronic specific heat is described by a power-law divergence, 𝐶el/𝑇∝𝑇^−0.54, aligning with previously observed ac-susceptibility and specific-heat measurements. The field dependence of the electronic specific heat at high magnetic fields shows a similar power law 𝐶el/𝑇∝𝐵^−0.50. In the zero-field and low-field region, we observe two small but distinct anomalies in the specific heat, located at 0.7 and 2.1 K.