Tunneling between opposite surfaces of a topological insulator thin film populated by electrons and holes is considered. We predict the considerable enhancement of tunneling conductivity by Cooper electron-hole pair fluctuations that are a precursor of their Cooper pairing. Cooper pair fluctuations lead to the critical behavior of tunneling conductivity in the vicinity of critical temperature with the critical index ν=2. If the pairing is suppressed by disorder, the behavior of tunneling conductivity in the vicinity of the quantum phase transition is also critical with the index μ=2. The effect can be interpreted as the fluctuational internal Josephson effect, and it is a general phenomenon for electron-hole bilayers. The peculiarities of the effect in other realizations of an electron-hole bilayer are discussed.