  Intergalactic medium temperature is a powerful probe of the epoch of
reionisation, as information is retained long after reionisation itself.
However, mean temperatures are highly degenerate with the timing of
reionisation, with the amount heat injected during the epoch, and with the
subsequent cooling rates. We post-process a suite of semi-analytic galaxy
formation models to characterise how different thermal statistics of the
intergalactic medium can be used to constrain reionisation. Temperature is
highly correlated with redshift of reionisation for a period of time after the
gas is heated. However as the gas cools, thermal memory of reionisation is
lost, and a power-law temperature-density relation is formed, $T =
T_0(1+\delta)^{1-\gamma}$ with $\gamma \approx 1.5$. Constraining our model
against observations of electron optical depth and temperature at mean density,
we find that reionisation likely finished at $z_{\rm{reion}} = 6.8 ^{+ 0.5}
_{-0.8}$ with a soft spectral slope of $\alpha = 2.8 ^{+ 1.2} _{-1.0}$. By
restricting spectral slope to the range $[0.5,2.5]$ motivated by population II
synthesis models, reionisation timing is further constrained to $z_{\rm{reion}}
= 6.9 ^{+ 0.4} _{-0.5}$. We find that, in the future, the degeneracies between
reionisation timing and background spectrum can be broken using the scatter in
temperatures and integrated thermal history.
