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We consider the non-unitary Lee-Yang minimal model ${\cal M}(2,5)$ in threedifferent finite geometries: (i) on the interval with integrable boundaryconditions labelled by the Kac labels $(r,s)=(1,1),(1,2)$, (ii) on the circlewith periodic boundary conditions and (iii) on the periodic circle including anintegrable purely transmitting defect. We apply $\varphi_{1,3}$ integrableperturbations on the boundary and on the defect and describe the flow of thespectrum. Adding a $\Phi_{1,3}$ integrable perturbation to move off-criticalityin the bulk, we determine the finite size spectrum of the massive scatteringtheory in the three geometries via Thermodynamic Bethe Ansatz (TBA) equations.We derive these integral equations for all excitations by solving, in thecontinuum scaling limit, the TBA functional equations satisfied by the transfermatrices of the associated $A_{4}$ RSOS lattice model of Forrester and Baxterin Regime III. The excitations are classified in terms of $(m,n)$ systems. Theexcited state TBA equations agree with the previously conjectured equations inthe boundary and periodic cases. In the defect case, new TBA equations confirmpreviously conjectured transmission factors.

Finite-Volume Spectra of the Lee-Yang Model