Let G = (V,E), V = {v1, v2, . . . , vn}, be a simple connected graph of order n and size m, without isolated vertices. Denote by d1 ≥ d2 ≥ · · · ≥ dn, di = d(vi) a sequence of vertex degrees of G. The general zeroth–order Randić index is defined as 0Rα(G) = ∑n i=1 dαi , where α is an arbitrary real number. The corresponding general zeroth–order Randić coindex is defined via 0Rα(G) = ∑n i=1(n − 1 − di)dαi . Some new bounds for the general zeroth–order Randić coindex and relationship between 0Rα(G) and 0Rα−1(G) are obtained. For a particular values of parameter α a number of new bounds for different topological coindices are obtained as corollaries.