The refined Gribov-Zwanziger framework takes into account the existence of equivalent gauge field configurations in the gauge-fixing quantization procedure of Euclidean Yang-Mills theories. Recently, this setup was extended to the family of linear covariant gauges giving rise to a local and Becchi-Rouet-Stora-Tyutin (BRST)-invariant action. In this paper, we give an algebraic proof of the renormalizability of the resulting action to all orders in perturbation theory.