The third law of thermodynamics states that the entropy of any system in equilibrium has to vanish at absolute zero temperature. At nonzero temperatures, on the other hand, matter is expected to accumulate entropy near a quantum critical point, where it undergoes a continuous transition from one ground state to another 1,2 . Here, we determine, based on general thermodynamic principles, the spatial-dimensional profile of the entropy S near a quantum critical point and its steepest descent in the corresponding multidimensional stress space. We demonstrate this approach for the canonical quantum critical compound CeCu 6−x Au x near its onset of antiferromagnetic order 2 . We are able to link the directional stress dependence of S to the previously determined geometry of quantum critical fluctuations 3 . Our demonstration of the multidimensional entropy landscape provides the foundation to understand how quantum criticality nucleates novel phases such as high-temperature superconductivity.Quantum criticality arises near a second-order phase transition that is driven to zero temperature by competing interactions. For metallic systems, it provides a mechanism to generate new types of electron-derived excitations that are distinct from Landau's Fermi liquid 2 . Because quantum fluctuations are enhanced when the dimensionality is reduced, quantum critical points (QCPs) often arise in anisotropic systems. The quantum critical fluctuations lead to unconventional scaling behaviour and the accumulation of entropy at very low T , thereby allowing unusual electronic excitations and new phases. The enhanced entropy S upon approaching a QCP has been probed by measurements of the specific heat, and its dependence on pressure was studied by volume thermal expansion. The entropy landscape has been studied up to now using a single tuning parameter 4 . To understand how entropy evolves as the system traverses near a QCP, exploration of its profile in a multidimensional parameter space is needed.Heavy-fermion systems represent prototype settings for QCPs induced by pressure. The latter tunes the hybridization of the almost localized 4f states with the conduction band, thereby tilting the balance in the competition between Ruderman-Kittel-Kasuya-Yosida (RKKY) and Kondo interactions. Previous experiments on quantum critical heavy-fermion systems 5,6 focused on the volume expansivity α V and volume Grüneisen ratio Γ V . Both α V /T and Γ V were found to diverge as T → 0, indicating a diverging pressure dependence of S and a vanishing energy scale near the QCP, as predicted 7 . Spatial anisotropy, a hallmark of many heavy-fermion systems, allows a QCP to be accessed with multiple tuning parameters.Here we show that, for anisotropic systems, the directional dependence of the thermal expansivity provides a means to determine the spatial-dimensional profile of the thermodynamic singularities near a QCP. We establish a procedure to identify the combination of stresses that aims directly at the QCP and accomplishes the steepest change of the...