  We calculate the fidelity susceptibility chi_f for the Luttinger model and
show that there is a universal contribution linear in temperature T (or inverse
length 1/L). Furthermore, we develop an algorithm - based on a lattice path
integral approach - to calculate the fidelity F(T) in the thermodynamic limit
for one-dimensional quantum systems. We check the Luttinger model predictions
by calculating chi_f(T) analytically for free spinless fermions and numerically
for the XXZ chain. Finally, we study chi_f at the two phase transitions in this
model.
