  We study the thermodynamic behaviour of the real $\beta$- and
$\gamma_i$-deformation of $\mathcal{N}=4$ Super Yang-Mills theory on
$\mathbb{R}\times S^3$ in the planar limit. These theories were shown to be the
most general asymptotically integrable supersymmetric and non-supersymmetric
field-theory deformations of $\mathcal{N}=4$ Super Yang-Mills theory,
respectively. We calculate the first loop correction to their partition
functions using an extension of the dilatation-operator and P\'{o}lya-counting
approach. In particular, we account for the one-loop finite-size effects which
occur for operators of length one and two. Remarkably, we find that the
$\mathcal{O}(\lambda)$ correction to the Hagedorn temperature is independent of
the deformation parameters, although the partition function depends on them in
a non-trivial way.
