This has to do with the fact that the analytic behaviour of the momentum-distribution function of electrons is of direct consequence to the decay property of various functions of wave vectors for large values of the amplitude of these vectors. For metals, where there is a discontinuity in the momentum-distribution function at the Fermi wave vectors, this decay is slowest (a power law). Roughly speaking, any non-vanishing temperature renders the momentum-distribution function smooth (infinitely differentiable) at the points of the underlying Fermi surface, leading to an exponential decay in the aforementioned functions at large values of the amplitude of wave vectors (there are some subtleties that I do not go into here, but are thoroughly discussed in my forthcoming paper as yet to be published). In practice, by choosing the temperature too small, the cross-over region where the decay of the relevant functions of wave vector change from power-law to exponential will be outside the adopted cut-off wave vector, implying that in practice there is a lower bound to the electronic temperature for it to contribute usefully to the convergence of the numerical results.