Propagation time through standard (solid core) optical fibers changes with temperature at a rate of 40 ps/km/K. The thermo-optic effect in silica glass accounts for about 95% of this change and thus hollow core fibers, in which the majority of optical power propagates through an air rather than glass core, can have this sensitivity greatly reduced. To date we have demonstrated a sensitivity as low as 2 ps/km/K, this value being limited by thermallyinduced fiber elongation. In this paper, we predict and experimentally demonstrate that the thermal sensitivity of propagation time can be reduced to zero (or even made negative) in Hollow Core Photonic Bandgap Fibers (HC-PBGF) by compensating the thermally-induced fiber elongation with an equal and opposite thermally-induced group velocity change (i.e. by making the light travel faster through the elongated fiber). This represents the ultimate fiber solution for many propagation time sensitive applications.