  We compute the specific heat $C(T)$ in an Fe-based superconductor with only
hole pockets. We use a three-orbital/three pocket model with two smaller hole
pockets made out of $d_{xz}$ and $d_{yz}$ orbitals and a larger pocket made out
of $d_{xy}$ orbital. We use as an input the experimental fact that the mass of
$d_{xy}$ fermion is much heavier than that of $d_{xz}/d_{yz}$ fermions. We
argue that the heavy $d_{xy}$ band contributes most to the specific heat in the
normal state, but the superconducting gap on the $d_{xy}$ pocket is
parametrically smaller than that on $d_{xz}/d_{yz}$ pockets. We argue that in
this situation the jump of $C(T)$ at $T_c$ is determined by $d_{xz}/d_{yz}$
fermions, and the ratio $(C_s-C_n)/C_n$ is a fraction of that in a one-band BCS
theory. Below $T_c$, $C(T)$ remains relatively flat down to some $T^*$, below
which $C(T)$ rapidly drops. This behavior is consistent with the data for
KFe$_2$As$_2$ and related materials. We argue that the data on $C(T)$ can be
reproduced without assuming that the quasiparticle residue $Z$ on $d_{xy}$ band
is small. We further argue that the very existence of a finite $T^* < T_c$
favors $s^{+-}$ gap structure over $d-$wave, because in the latter case $T^*$
would vanish.
