Effect of pressure on transition temperature
The mathematical answer is based on Clapeyron's equation (written in what follows in form inverted relative to the more usual):
$$\frac{dT}{dp}= \frac{\Delta V}{\Delta S}$$
which must hold on any point on the phase coexistence line.
If $\frac{dT}{dp}=0$ then it follows that $\Delta V=0$.