We study the many-body physics in thin-film topological band insulator, where the inter-edge Coulomb interaction can lead to an exciton condensation transition. We investigate the universality class of the exciton condensation quantum critical point. With different chemical potentials and interactions, the exciton condensation can belong to z = 2 mean field, or 3d XY, or Yukawa-Higgs universality classes. The interplay between exciton condensate and the time-reversal symmetry breaking is also discussed. Predictions of our work can be tested experimentally by tuning the chemical potentials on both surfaces of the thin film through gate voltage. We also show that all the analysis of the exciton condensate can be directly applied to a spin-triplet superconductor phase with attractive inter-edge interaction.