  The pyrochlore lattice is associated with several potential and actual spin
liquid phases as a result of its strong geometric frustration. At finite
temperature, these can exhibit an unusually broad cross-over regime to a
conventional paramagnet. Here, we study this regime analytically by showing how
a single-tetrahedron Hamiltonian can extrapolate beyond the first term of a
high-temperature expansion and yield insights into the build-up of
correlations. We discuss how this unusual behaviour is brought about by the
structure of the eigenspaces of the coupling matrix. Further interesting
behaviour can appear for parameter values located near phase transitions: we
find coexistence of $(111)$ rods and $(220)$ peaks in the structure factor, as
observed in neutron scattering experiments on Yb$_2$Ti$_2$O$_7$.
