  There is considerable evidence that the superconductivity of Sr2RuO4 has two
components. Among this evidence is a jump in the shear elastic modulus $c_{66}$
at the critical temperature $T_c$, observed in ultrasound measurements. Such a
jump is forbidden for homogeneous single-component order parameters, and
implies that $T_c$ should develop as a cusp under the application of shear
strain with $\langle 110 \rangle$ principal axes. This shear strain should
split the onset temperatures of the two components, if they coexist, or select
one component if they do not. Here, we report measurements of $T_c$ and the
elastocaloric effect of Sr2RuO4 under uniaxial stress applied along the $[110]$
lattice direction. Within experimental resolution, we resolve neither a cusp in
the stress dependence of $T_c$, nor any second transition in the elastocaloric
effect data. We show that reconciling these null results with the observed
jumps in $c_{66}$ requires extraordinarily fine tuning to a triple point of the
Ginzburg-Landau parameter space. In addition, our results are inconsistent with
homogeneous time reversal symmetry breaking at a temperature $T_2 \leq T_c$ as
identified in muon spin relaxation experiments.
