Despite the success in describing a range of quantum many-body states using tensor networks, there is a no-go theorem that rules out strictly local tensor networks as topologically nontrivial groundstates of gapped parent Hamiltonians with short-range (including exponentially decaying) couplings. In this work, we show that for free fermions, strictly local tensor networks may describe nonzero temperature averages with respect to gapped Hamiltonians with exponentially decaying couplings. Parent Hamiltonians in this sense may be constructed for any dimensionality and without any obstructions due to their topology. Conversely, we also show that thermal averages with respect to gapped, strictly short-range free-fermion Hamiltonians can be calculated by tensor networks whose links decay exponentially with distance. We also describe a truncation-reconstruction scheme for such tensor networks that leads to a controlled approximation of exact averages in terms of a sequence of related thermal averages. We illustrate our scheme on the two-dimensional Haldane honeycomb model considering both topological and nontopological phases.