  We present a thorough study of the thermodynamics of a one-dimensional
repulsive Bose gas, focusing in particular on corrections beyond the
Luttinger-liquid description. We compute the chemical potential, the pressure
and the contact, as a function of temperature and gas parameter with exact
thermal Bethe-Ansatz. In addition, we provide interpretations of the main
features in the analytically tractable regimes, based on a variety of
approaches (Bogoliubov, hard-core, Sommerfeld and virial). The beyond
Luttinger-liquid thermodynamic effects are found to be non-monotonic as a
function of gas parameter. Such behavior is explained in terms of non-linear
dispersion and ``negative excluded volume'' effects, for weak and strong
repulsion respectively, responsible for the opposite sign corrections in the
thermal next-to-leading term of the thermodynamic quantities at low
temperatures. Our predictions can be applied to other systems including super
Tonks-Girardeau gases, dipolar and Rydberg atoms, helium, quantum liquid
droplets in bosonic mixtures and impurities in a quantum bath.
