  At high temperature, the overlap of two particles chosen independently
according to the Gibbs measure of the branching Brownian motion converges to
zero as time goes to infinity. We investigate the precise decay rate of the
probability to obtain an overlap greater than $a$, for some $a>0$, in the whole
subcritical phase of inverse temperatures $\beta \in [0,\beta_c)$. Moreover, we
study this probability both conditionally on the branching Brownian motion and
non-conditionally. Two sub-phases of inverse temperatures appear, but
surprisingly the threshold is not the same in both cases.
