  Many ceramic and hybrid metal-organic framework materials show negative
thermal expansion (NTE): they \textit{contract} instead of expanding on heating
\cite{Barrera_Miller_Lind_Romao 2005}. Their structures are invariably
characterised as a network of polyhedral groups of atoms that are connected
through sharing of corner atoms or by shared ligands. Empirically, NTE
materials tend to show pressure-induced softening, pressure enhancement of NTE,
and the reduction of NTE on heating. But such effects have only been
investigated in a small number of materials \cite{Pantea 2006,Chapman
2005,Chapman 2007,Fangexp 2013}, and as yet there is no general framework for
understanding the whole suite of properties together. By studying models with
Hamiltonians chosen to reflect the physical picture generally accepted as
responsible for NTE in framework materials, we demonstrate that NTE,
pressure-enhanced NTE, and pressure-induced softening naturally emerge
together. We then show how anharmonic interactions lead to structural warm
hardening---something that has only previously been seen in laser-excited
warm-dense matter \cite{Ernstorfer 2009}---as well as to the transition from
NTE to positive thermal expansion and the disappearing of the pressure-induced
softening at high temperatures.
