Significance Research on the unconventional superconductivity of S r 2 R u O 4 is undergoing a renaissance since recent spin susceptibility measurements ruling out the spin triplet order parameter which had been widely favored for over two decades. With ultrasound, Kerr rotation, and muon spin relaxation data all providing evidence for a two-component order parameter, it is vital that this possibility be investigated thermodynamically by studying the dependence of the heat-capacity anomaly on uniaxial pressure. Here, the relevant experimental results are combined with theoretical analysis that shows how strongly the data constrain theories of the order parameter. In particular, we do not observe any signs of transition splitting of two-order-parameter components. S r 2 R u O 4 thus offers a unique test bed for theories of unconventional superconductivity.