  Inspired by recent results in the context of AdS/CFT integrability, we
reconsider the Thermodynamic Bethe Ansatz equations describing the 1D fermionic
Hubbard model at finite temperature. We prove that the infinite set of TBA
equations are equivalent to a simple nonlinear Riemann-Hilbert problem for a
finite number of unknown functions. The latter can be transformed into a set of
three coupled nonlinear integral equations defined over a finite support, which
can be easily solved numerically. We discuss the emergence of an exact Bethe
Ansatz and the link between the TBA approach and the results by J\"uttner,
Kl\"umper and Suzuki based on the Quantum Transfer Matrix method. We also
comment on the analytic continuation mechanism leading to excited states and on
the mirror equations describing the finite-size Hubbard model with twisted
boundary conditions.
