A key question regarding the unconventional superconductivity of Sr$_2$RuO$_4$ remains whether the order parameter is single- or two-component. Under a hypothesis of two-component superconductivity, uniaxial pressure is expected to lift their degeneracy, resulting in a split transition. The most direct and fundamental probe of a split transition is heat capacity. Here, we report development of new high-frequency methodology for measurement of heat capacity of samples subject to large and highly homogeneous uniaxial pressure. We place an upper limit on the heat capacity signature of any second transition of a few per cent of the primary superconducting transition. The normalized jump in heat capacity, $Delta C/C$, grows smoothly as a function of uniaxial pressure, but we find no qualitative evidence of a pressure-induced order parameter transition. Thanks to the high precision of our measurements, these findings place stringent constraints on theories of the superconductivity of Sr$_2$RuO$_4$.
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