We study the rapidity distribution in the Lieb-Liniger model and derive exact relations for its derivatives at the Fermi level. The latter enables us to treat analytically the free energy of the system at low temperatures and arbitrary interactions. We calculated the leading-order correction to the well-known result obtained using conformal field theory. In contrast to the leading-order term controlled by the sound velocity or the Luttinger liquid parameter, the new term is controlled by an additional dimensionless parameter. We calculated its series expansions in the limiting cases of weak and strong interactions. Our results are generalized to other Galilean-invariant integrable systems.