Single crystal Cu(001) layers were grown on MgO(001) by ultrahigh vacuum magnetron sputtering at Ts=100 °C. Quantitative surface morphological analyses by in situ scanning tunneling microscopy show that the surfaces exhibit self-affine mound structures with a scaling exponent of 0.82±0.03 and a mound radius rc that increases from 31±8 to 39±6 nm for increasing layer thickness t=24–120 nm. In situ annealing at 200 and 300 °C leads to a thermodynamically driven mass transport that minimizes the surface step density, resulting in broader mounds and a smaller root mean square surface roughness σ. This effect is most pronounced for t=24 nm, for which rc increases from 31±8 to 70±20 nm and σ decreases from 1.3±0.1 to 0.74±0.08 nm, resulting in a decrease in the average surface slope from χ=7° to 2° and an increase in the average terrace width wT by more than a factor of 4. In contrast, wT increases by only 20% for t=120 nm. This remarkable difference between “thin” and “thick” layers is attributed to diverging surface morphological pathways during annealing: The strong smoothening for t=24 nm is due to a competitive coalescence process where some mounds grow laterally at the expense of their smaller neighbors, which die o- ut. In contrast, the initially wider mounds of thicker layers (t=120 nm) combine to form a quasistable surface morphology that exhibits anisotropic mound structures, which limit mass transport and stabilize the surface step density.