In this paper we consider double cosine series whose coefficients form a null sequence of bounded variation of order $(p, 0)$, $(0, p)$ and $(p, p)$ with the weight $(jk)^{p-1}$ for some $p> 1$. We study pointwise convergence, uniform convergence and convergence in $L^r$-norm of the series under consideration. In a certain sense our results extend the results of Young \cite{Young}, Kolmogorov \cite{Kolmogorov} and Móricz \cite{Moricz1,Moricz2}.