  We study the spectral problem in deformed supersymmetric quantum mechanics
with polynomial superpotential by using the exact WKB method and the TBA
equations. We apply the ODE/IM correspondence to the Schr\"odinger equation
with an effective potential deformed by integrating out the fermions, which
admits a continuous deformation parameter. We find that the TBA equations are
described by the ${\mathbb Z}_4$-extended ones. For cubic superpotential
corresponding to the symmetric double-well potential, the TBA system splits
into the two $D_3$-type TBA equations. We investigate in detail this example
based on the TBA equations and their analytic continuation as well as the
massless limit. We find that the energy spectrum obtained from the exact
quantization condition is in good agreement with the diagonalization approach
of the Hamiltonian.
