ABSTRACT A strong correlation between speed and proton temperature has been observed, across many years, on hourly averaged measurements in the solar wind. Here, we show that this relationship is also observed at a smaller scale on intervals of a few days, within a single stream. Following the radial evolution of a well-defined stream of coronal-hole plasma, we show that the temperature–speed (T–V) relationship evolves with distance, implying that the T–V relationship at 1 au cannot be used as a proxy for that near the Sun. We suggest that this behaviour could be a combination of the anticorrelation between speed and flux-tube expansion factor near the Sun and the effect of a continuous heating experienced by the plasma during the expansion. We also show that the cooling index for the radial evolution of the temperature is a function of the speed. In particular, T⊥ in faster wind, although higher close to the Sun, decreases more quickly with respect to slower wind, suggesting that it has less time to interact with the mechanism(s) able to heat the plasma. Finally, we predict the expected T–V relationship in fast streams closer to the Sun with respect to the Helios observations, which Parker Solar Probe will explore in the near future.