We report the detailed analysis of specific heat [C$_{rm P}$(T)] and ac-susceptibility for magnetically frustrated Sr$_{2-x}$La$_x$CoNbO$_6$ ($x=$ 0--1) double perovskites to understand low temperature complex magnetic interactions and their evolution with $x$. Interestingly, the observed Schottky anomaly in the $xleqslant$ 0.4 samples shifts gradually towards higher temperature with magnetic field as well as $x$, and the analysis reveal the persistence of the discrete energy states in these samples resulting from the spin-orbit coupling and octahedral distortion. Moreover, the extracted values of Lande g--factor indicate the existence of high-spin state Co$^{3+}$ ions close to non-magnetic low-spin state. The specific heat data show the $lambda$-type anomaly for the $xgeqslant$ 0.6 samples due to evolution of the long range antiferromagnetic ordering. Our analysis of low temperature C$_{rm P}$(T) data for the $xgeqslant$ 0.6 samples demonstrate the 3D isotropic Heisenberg antiferromagnetic (AFM) interactions and the temperature induced second order AFM--paramagnetic phase transition. More interestingly, we demonstrate the presence of the free Co$^{2+}$ like Kramers doublet ground state in the $x =$1 sample. Further, the ac susceptibility and time evolution of the magnetization data reveal the low temperature cluster-glass like behavior in the $x=$ 0--0.4 samples, where spin-spin correlation strength decreases with $x$.
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