  The cuprate superconductor YBa$_2$Cu$_4$O$_8$, in comparison with most other
cuprates, has a stable stoichiometry, is largely free of defects and may be
regarded as the canonical underdoped cuprate, displaying marked pseudogap
behaviour and an associated distinct weakening of superconducting properties.
This cuprate `pseudogap' manifests as a partial gap in the electronic density
of states at the Fermi level and is observed in most spectroscopic properties.
After several decades of intensive study it is widely believed that the
pseudogap closes, mean-field like, near a characteristic temperature, $T^*$,
which rises with decreasing hole concentration, $p$. Here, we report extensive
field-dependent electronic specific heat studies on YBa$_2$Cu$_4$O$_8$ up to an
unprecedented 400 K and show unequivocally that the pseudogap never closes,
remaining open to at least 400 K where $T^*$ is typically presumed to be about
150 K. We show from the NMR Knight shift and the electronic entropy that the
Wilson ratio is numerically consistent with a weakly-interacting Fermion system
for the near-nodal states. And, from the field-dependent specific heat, we
characterise the impact of fluctuations and impurity scattering on the
thermodynamic properties.
