  We use a combination of variable-temperature / variable-pressure neutron
powder diffraction, variable-pressure inelastic neutron scattering, and quantum
chemical calculations to interrogate the behaviour of the negative thermal
expansion (NTE) material $^{114}$Cd(CN)$_2$ under hydrostatic pressure. We
determine the equation of state of the ambient-pressure phase, and discover the
so-called `warm hardening' effect whereby the material becomes elastically
stiffer as it is heated. We also identify a number of high-pressure phases, and
map out the phase behaviour of Cd(CN)$_2$ over the range $0\leq p\leq0.5$\,GPa,
$100\leq T\leq300$\,K. As expected for an NTE material, the low-energy phonon
frequencies are found to soften under pressure, and we determine an effective
Gr{\"u}neisen parameter for these modes. Finally, we show that the elastic
behaviour of Cd(CN)$_2$ is sensitive to the local Cd coordination environment,
which suggests an interplay between short- (phononic) and long-timescale
(cyanide flips) fluctuations in Cd(CN)$_2$.
