For graphs G and H, let G⊕H denote their Cartesian sum. We investigate the chromatic number and the circular chromatic number for G⊕H. It has been proved that for any graphs G and H, χ(G⊕H)≤max{⌈χc(G)χ(H)⌉,⌈χ(G)χc(H)⌉}. It has been conjectured that for any graphs G and H, χc(G⊕H)≤max{χ(H)χc(G),χ(G)χc(H)}. We confirm this conjecture for graphs G and H with special values of χc(G) and χc(H). These results improve previously known bounds on the corresponding coloring parameters for the Cartesian sum of graphs.