Critical surface band gap of repulsive Casimir interaction between three dimensional topological insulators at finite temperature

We generalize the calculation of Casimir interaction between topological insulators with opposite topological magnetoelectric polarizabilities and finite surface band gaps to finite Temperature cases. We find that finite temperature quantitatively depress the repulsive peak and enlarge the critical surface gap $m_c$ for repulsive Casimir force. However the universal property $m_c a \sim 1/2$ is still valid for various oscillation strength, temperature region and topological magnetoelectric polarizabilities.