  Since its experimental discovery, many phenomenological theories successfully
reproduced the rapid rise from $p$ to $1+p$ found in the Hall number $n_H$ at
the critical doping $p^*$ of the pseudogap in superconducting cuprates. Further
comparison with experiments is now needed in order to narrow down candidates.
In this paper, we consider three previously successful phenomenological
theories in a unified formalism---an antiferromagnetic mean field (AF), a
spiral incommensurate antiferromagnetic mean field (sAF), and the
Yang-Rice-Zhang (YRZ) theory. We find a rapid rise in the specific heat and a
rapid drop in the Seebeck coefficient for increasing doping across the
transition in each of those models. The predicted rises and drops are locked,
not to~$p^*$, but to the doping where anti-nodal electron pockets,
characteristic of each model, appear at the Fermi surface shortly before~$p^*$.
While such electron pockets are still to be found in experiments, we discuss
how they could provide distinctive signatures for each model. We also show that
the range of doping where those electron pockets would be found is strongly
affected by the position of the van~Hove singularity.
