Thermotropic liquid crystalline polymers commonly develop a bubble texture at time scales necessary to perform low‐rate rheological experiments. We have fabricated a capillary rheometer capable of high‐sensitivity (relatively low‐rate) measurements under a hydrostatic pressure adequate to suppress formation of visible bubbles. Vectra A (a copolyester of 73% HBA and 27% HNA) exhibited a transient reduction in viscosity at low rates that is independent of strain history and dependent only on time in the melt; no change in inherent viscosity was observed despite a threefold change in viscosity. The transient, which was not observed at high rates on the time scale of the experiment, may reflect changes in the defect structure. The flow curve for a given thermal history (time in the melt) exhibited a flattening at low to intermediate rates, suggestive of the flow curves reported for lyotropic systems. This result is qualitatively in agreement with rotational rheometer measurements known to be contaminated by offgassing, which exhibit higher apparent viscosities but give the same flow curve shape. Rotational rheometer transients scaled with strain in a manner similar to that reported for lyotropic systems. Transient reduction in viscosity and flow curve flattening with decreasing rate was also observed in pressurized rheometer measurements of the fully nematic copolyester of 80% HBA and 20% PET; a decrease in inherent viscosity was observed for this polymer, but it appears to be too small to account for the observed decrease in viscosity with time.