Think of the equilibrium constant Ksol. We know (I hope) that lnKsol = -ΔGsol°/RT = -ΔHsol°/RT + ΔSsol°/R
Thus dlnKsol/dT = ΔHsol°/RT2 (assuming ΔH and ΔS are constant with temperature, which is usually true to a first approximation, over a relatively small temperature range)
The RHS is fairly constant over a small temperature range, hence it is lnK, not K, that varies linearly with temperature.