  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 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 that of the primary
superconducting transition. The normalized jump in heat capacity, $\Delta C/C$,
grows smoothly as a function of uniaxial pressure, favouring order parameters
which are allowed to maximize in the same part of the Brillouin zone as the
well-studied van Hove singularity. Thanks to the high precision of our
measurements, these findings place stringent constraints on theories of the
superconductivity of Sr$_2$RuO$_4$.
