  Explicit expressions for quantum fluctuations of energy in subsystems of a
hot relativistic gas of spin-$1/2$ particles are derived. The results depend on
the form of the energy-momentum tensor used in the calculations, which is a
feature described as pseudo-gauge dependence. However, for sufficiently large
subsystems the results obtained in different pseudo-gauges converge and agree
with the canonical-ensemble formula known from statistical physics. As
different forms of the energy-momentum tensor of a gas are a priori equivalent,
our finding suggests that the concept of quantum fluctuations of energy in very
small thermodynamic systems is pseudo-gauge dependent. On the practical side,
the results of our calculations determine a scale of coarse graining for which
the choice of the pseudo-gauge becomes irrelevant.
