  We consider a coarse-grained polymer model in order to investigate the origin
of a recently discovered negative energy-related contribution to the elastic
modulus $G(T)$ of rubber-like gels. From this model, we are able to compute an
exact expression for the free energy of the system, which allows us to evaluate
a stress-strain relationship that displays a non-trivial dependence on the
temperature $T$. We validate our approach through comparisons between the
theoretical results and the experimental data obtained for tetra-PEG hydrogels,
which indicate that, although simple, the present model works well to describe
the experiments. Importantly, our approach unveiled aspects of the experimental
analysis which turned out to be different from the conventional entropic and
energetic analysis broadly used in the literature. Also, in contrast to the
linear dependence predicted by the traditional, {\it i.e.}, purely entropic,
models, our results suggest that the general expression of the elastic modulus
should be of the form $G(T) \propto k_BT w(T)$, with $w(T)$ being a
temperature-dependent correction factor that could be related to the
interaction between the chains in the network and the solvent. Accordingly, the
correction factor allows the expression found for the elastic modulus to
describe both rubber and rubber-like gels.
