  We consider the 3D or 2D primitive equations for oceans and atmosphere in the
isothermal setting. In this paper, we establish a new conditional uniqueness
result for weak solutions to the primitive equations, that is, if a weak
solution belongs some scaling invariant function spaces, and satisfies some
additional assumptions, then the weak solution is unique. In particular, our
result can be obtained as different one from $z$-weak solutions framework by
adopting some anisotropic approaches with the homogeneous toroidal Besov
spaces. As an application of the proof, we establish the energy equality for
weak solutions in the uniqueness class given in the main theorem.
