The diffusive transport rate in a soft matter nanofluidic device is controlled with a thermoactuated hydrogel valve. The device consists of three giant unilamellar vesicles linearly conjugated by lipid nanotubes, with a solution of the stimuli-responsive polymer poly (N-isopropyl acrylamide) (PNIPAAm) in the central vesicle. The valve states "high (transport) rate" and "low (transport) rate" are obtained by heat-activated switching between PNIPAAm's dissolved and compact aggregated states. We show that three parameters influence the diffusion rate within the device: the increase of the transport rate caused by a decrease in PNIPAAm concentration upon compaction, the temperature dependence of the buffer viscosity, and the volume excluded by the PNIPAAm hydrogel compartment.