  Recent high-resolution cosmological hydrodynamic simulations run with a
variety of codes systematically predict large amounts of entropy in the
intra-cluster medium at low redshift, leading to flat entropy profiles and a
suppressed cool-core population. This prediction is at odds with X-ray
observations of groups and clusters. We use a new implementation of the EAGLE
galaxy formation model to investigate the sensitivity of the central entropy
and the shape of the profiles to changes in the sub-grid model applied to a
suite of zoom-in cosmological simulations of a group of mass $M_{500} = 8.8
\times 10^{12}~{\rm M}_\odot$ and a cluster of mass $2.9 \times 10^{14}~{\rm
M}_\odot$. Using our reference model, calibrated to match the stellar mass
function of field galaxies, we confirm that our simulated groups and clusters
contain hot gas with too high entropy in their cores. Additional simulations
run without artificial conduction, metal cooling or AGN feedback produce lower
entropy levels but still fail to reproduce observed profiles. Conversely, the
two objects run without supernova feedback show a significant entropy increase
which can be attributed to excessive cooling and star formation. Varying the
AGN heating temperature does not greatly affect the profile shape, but only the
overall normalisation. Finally, we compared runs with four AGN heating schemes
and obtained similar profiles, with the exception of bipolar AGN heating, which
produces a higher and more uniform entropy distribution. Our study leaves open
the question of whether the entropy core problem in simulations, and
particularly the lack of power-law cool-core profiles, arise from incorrect
physical assumptions, missing physical processes, or insufficient numerical
resolution.
