  We derive the Thermodynamic Bethe Ansatz (TBA) equations for the
Schr\"odinger equation with an arbitrary polynomial potential and a regular
singular (simple and double pole) term. The TBA equations provide a non-trivial
generalization of the ODE/IM correspondence and also give a solution for the
Riemann-Hilbert problem in the exact WKB method. We study the TBA equations in
detail for the linear and the harmonic oscillator potentials together with
inverse and centrifugal terms. As an application, we also compute numerically
the Voros spectrum for these potentials using the Bohr-Sommerfeld quantization
condition.
