We investigate the low-temperature phase diagram of the exactly solved su(4) two-leg spin ladder as a function of the rung coupling J⊥ and magnetic field H by means of the thermodynamic Bethe ansatz (TBA). In the absence of a magnetic field the model exhibits three quantum phases, while in the presence of a strong magnetic field there is no singlet ground state for ferromagnetic rung coupling. For antiferromagnetic rung coupling, there is a gapped phase in the regime H < Hc1, a fully polarized gapped phase for H > Hc2 and a Luttinger liquid magnetic phase in the regime Hc1 < H < Hc2. The critical behaviour derived using the TBA is consistent with the existing experimental, numerical and perturbative results for the strong coupling ladder compounds. This includes the spin excitation gap and the critical fields Hc1 and Hc2, which are in excellent agreement with the experimental values for the known strong coupling ladder compounds (5IAP)2CuBr4·2H2O, Cu2(C5H12N2)2Cl4 and (C5H12N)2CuBr4. In addition we predict the spin gap ∆ ≈ J⊥−1/2J║ for the weak coupling compounds with J⊥ ∼ J║ , such as (VO)2P2O7, and also show that the gap opens for arbitrary J⊥/J║ .