In this paper we present a study of the connectivity degrees of the threshold preferential attachment model, a generalization of the Barabási-Albert model to heterogeneous complex networks. The threshold model incorporates the states of the nodes in its preferential linking rule and assumes that the affinity between network nodes follows an inverse relationship with the distance between their states. We numerically analyze the connectivity degrees of the model, studying the influence of the main parameters on the distribution of connectivity degrees and its statistics, the average degree and highest degree of the network. We show that such statistics exhibit markedly different behaviors in the dependence on the model parameters, particularly as regards the interaction threshold. Nevertheless, we show that the two statistics converge in the limit of null threshold and often exhibit scaling that can be described by power laws of the model parameters.