We investigate the electronic specific heat of overdoped BaFe$_{2}$(As$_{1-x}$P$_{x}$)$_{2}$ single crystals in the superconducting state using high-resolution nanocalorimetry. From the measurements, we extract the doping dependence of the condensation energy, superconducting gap $Delta$, and related microscopic parameters. We find that the anomalous scaling of the specific heat jump $Delta C propto T_{mathrm{c}}^3$, found in many iron-based superconductors, in this system originates from a $T_mathrm{c}$-dependent ratio $Delta/k_mathrm{B}T_mathrm{c}$ in combination with a doping-dependent density of states $N(varepsilon_mathrm{F})$. A clear enhancement is seen in the effective mass $m^{*}$ as the composition approaches the value that has been associated with a quantum critical point at optimum doping. However, a simultaneous increase in the superconducting carrier concentration $n_mathrm{s}$ maintains the superfluid density, yielding an apparent penetration depth $lambda$ that decreases with increasing $T_mathrm{c}$ without sharp divergence at the quantum critical point. Uemura scaling indicates that $T_mathrm{c}$ is governed by the Fermi temperature $T_mathrm{F}$ for this multi-band system.
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