  We report an angle-resolved photoemission spectroscopy (ARPES) study of
KFe$_2$As$_2$ and CsFe$_2$As$_2$, revealing the existence of a van Hove
singularity affecting the electronic properties. As a result of chemical
pressure, we find a stronger three-dimensionality in KFe$_2$As$_2$ than in
CsFe$_2$As$_2$, notably for the 3$d_{z^2}$ states responsible for the small
three-dimensional hole-like Fermi surface pocket reported by quantum
oscillations. Supported by first-principles calculations, our ARPES study shows
that the van Hove singularity previously reported in KFe$_2$As$_2$ moves closer
to the Fermi level under negative chemical pressure. This observation, which
suggests that the large density-of-states accompanying the van Hove singularity
contributes to the large Sommerfeld coefficient reported for the AFe$_2$As$_2$
(A = K, Rb, Cs) series, is also consistent with the evolution of the inelastic
scattering revealed by transport under external pressure, thus offering a
possible interpretation for the origin of the apparent change in the
superconducting order parameter under pressure. We find that the coherent
spectral weight decreases exponentially upon increasing temperature with a
characteristic temperature $T^*$. We speculate how the low-energy location of
the van Hove singularity and the presence of a low-energy peak in the phonon
density-of-states can relate to the high-temperature crossover observed in
various electronic and thermodynamic quantities.
