5 SUPPLEMENTARY MATERIAL • Evaluation of the normal-state specific heat at low temperature In order to reliably extract the electronic contribution, C$_{el}$ ( T ) , from the total measured specific heat, the contribution of lattice excitations, C$_{latt}$ ( T ) , has to be accurately determined. The conventional method to evaluate the lattice specific heat of a superconductor is to recover the normal state by applying a magnetic field. This is not feasible for Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ , because the requisite field range is experimentally inaccessible. In our work, by using the lattice specific heat of Ba(Fe$_{0}$$_{.}$$_{88}$Mn$_{0}$$_{.}$$_{12}$)$_{2}$As$_{2}$ , C Mn $_{latt}$( T ) , as a reference, we evaluated C K $_{latt}$( T ) of Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ as A · C Mn $_{latt}$( B · T ) , with scaling factors A and B of 0.95 and 1.03, respectively. The green curve in Fig. S1 represents the resulting normal-state specific heat of Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ from a least-squares fit of our data (open circles) to Eq.(S1) within the temperature range 40 - 160 K under the constraint of entropy conservation. Alternatively, we also simulated C K $_{latt}$( T ) by representing phonon density of states (PDOS) F ( ω ) by six δ -functions F$_{i}$δ ( ω − ω$_{i}$ ) on a basis of Einstein frequencies in a geometrical ratio ω$_{i}$$_{+1}$ = 1 . 75 · ω$_{i}$ . The corresponding normal-state specific heat is given by where R is the universal gas constant, x$_{i}$ = $_{ℏ}$ω$_{i}$ /k$_{B}$T , and the sum of the weights F$_{i}$ is constrained to the number of atoms per unit cell. A justification of this trial PDOS is addressed, e.g., in Refs. S1-S3. The number of modes was chosen to be small enough to ensure the stability of the solution. The geometric ratio 1.75:1 allowed us to avoid spurious oscillations of the phonon DOS. The weights F$_{i}$ , the starting Einstein frequency ω$_{1}$ , and the Sommerfeld constant γ N K ( ≈ 49 mJ/molK $^{2}$) were determined by a least-squares fit of the high temperature (40 K < T < 200 K) specific heat of Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ (open circles in Fig. S2) to Eq.(S1) under the constraint of entropy conservation. The results of the fit is given in Table SI. The δ -functions of the phonon DOS F ( ω ) are represented by a histogram in the inset of Fig. S2. The frequencies $_{ℏ}$ω$_{i}$ are ranging from 2 to 32 meV with the C K $_{tot}$( T ) = C K $_{latt}$( T ) + γ N K · T = 3 R 6 ∑ i =1 F$_{i}$ x 2 $_{i}$e x$_{i}$ ( e x$_{i}$ − 1) 2 + γ N K · T, 6 ∑ i =1 F$_{i}$ = 5 (S1) FIG. S2: Temperature dependence of (circles) the specific heat C$_{p}$ ( T ) /T of Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ and (blue curve) its normal-state counterpart resulting from the simulation of the lattice specific heat on a basis of Einstein modes, after Eq. (S1). Inset: The δ -functions F$_{i}$δ ( ω − ω$_{i}$ ) of the PDOS used for the fit are presented by a histogram. FIG. S1: Temperature dependence of (circles) the specific heat C$_{p}$ ( T ) /T of Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ and (green curve) its normal-state counterpart resulting from the corresponding state approximation by using the lattice specific heat of Ba(Fe$_{0}$$_{.}$$_{88}$Mn$_{0}$$_{.}$$_{12}$)$_{2}$As$_{2}$ as a refer- ence, after Eq. (1) of the Letter. TABLE SI: Frequencies and weights of the six Einstein modes fit to the lattice specific heat of Ba$_{0}$$_{.}$$_{68}$K$_{0}$$_{.}$$_{32}$Fe$_{2}$As$_{2}$ ( γ = 49 mJ/molK $^{2}$). i $_{ℏ}$ω$_{i}$ , meV (K) F$_{i}$ 1 1.92 (22.3) 0.005 2 3.36 (39.0) 0.018 3 5.89 (68.4) 0.136 4 10.30 (119.6) 1.380 5 18.04 (209.4) 1.450 6 31.57 (366.4) 2.020