  Recent experiments in unconventional superconductors, and in particular
iron-based materials, have reported evidence of an antiferromagnetic quantum
critical point (AFM-QCP) emerging inside the superconducting dome of the phase
diagram. Fluctuations associated with such an AFM-QCP are expected to promote
unusual temperature dependencies of thermodynamic quantities. Here, we compute
the $T$ dependence of the specific heat $C(T)$ deep inside a fully gapped
$s^{+-}$ superconducting state as the AFM-QCP is approached. We find that, at
the AFM-QCP, the specific heat $C(T)$ vanishes quadratically with temperature,
as opposed to the typical exponential suppression seen in fully-gapped BCS
superconductors. This robust result is due to a non-analytic contribution to
the free-energy arising from the general form of the bosonic (AFM) propagator
in the SC state. Away from the AFM-QCP, as temperature is lowered, $C(T)$ shows
a crossover from a $T^2$ behavior to an exponential behavior, with the
crossover temperature scale set by the value of the superconducting gap and the
distance to the QCP. We argue that these features in the specific heat can be
used to unambiguously determine the existence of AFM-QCPs inside the
superconducting domes of iron-based and other fully gapped unconventional
superconductors.
