It is known that 12-hydroxystearic acid coupled via ion-pairing with a counterion, such as ethanolamine, self-assembles into supramolecular multilayer tubes that exhibit a peculiar temperature-tunable diameter variation. At a given temperature threshold (T(phi t)), there is a large increase of the tube diameter by almost a factor of 10. We investigate here the nature of the counterion on the self-assembly of such tubes and the variation of their diameter with the temperature. Eight different counterions are used to produce tubes via ion-pairing with 12-hydroxystearic acid. We systematically measure structural parameters at different scales coupling phase contrast microscopy and small-angle neutron scattering (SANS), in combination with thermodynamic studies using differential scanning calorimetry (DSC). The temperature T(phi t) is found to depend sensitively on the nature of the counterion. One could not identify any DSC peak associated with the diameter variation. However, a detailed line-shape analysis of the SANS spectra reveals that a significant softening of the elastic properties of the multilayer tube walls occurs at T(phi t), for all counterions investigated. Depending on both the temperature and the nature of the counterion, the interlayer spacing in the multilayered structure (respectively, the bilayer thickness) varies in a range from 240 to 440 angstrom (respectively, from 22 to 42 angstrom), but those variations appear not to be related to T(phi t). Altogether, our results show that the variation of the diameter is correlated to the nature of the counterion and to the elastic properties of the bilayer stack.