  We derive the path integral of the semiclassical, one dimensional anharmonic
Holstein model assuming that the electron motion takes place in a bath of non
linear oscillators with quartic on site hard (and soft) potentials. The
interplay between {\it e-ph} coupling and anharmonic force constant is analysed
both in the adiabatic and antiadiabatic regime. In the latter we find much
larger anharmonic features on the thermodynamic properties of low energy
oscillators. Soft on site potentials generate attractive centres at large
amplitude oscillator paths and contribute to the anomalous shape of the {\it
heat capacity over temperature} ratio in the intermediate to low $T$ range.
This anharmonic lattice effect is superimposed to the purely electronic
contribution associated to a temperature dependent hopping with variable range
inducing local disorder in the system.
