We report the temperature(T)-dependence of the planar Cu-63 nuclear spin-lattice relaxation rate, 1/T-1, both at the Cu(A)-site (77.3 K less than or equal to T less than or equal to 500 K) and at the Cu(B)-site (77.3 K less than or equal to T less than or equal to 300 K) in the high-T-c Cu oxide, La1.80Sr0.20CuO4 (T-c similar to 30 K), measured by the Cu NQR spin-echo technique. We found that 1/T-1 at the Cu(B)-site is different from that at the Cu(A)-site, and that 1/T-1 at the Cu(A)-site has a superlinear T-dependence in the T-range of 300 similar to 500 K. Comparisons of the present 1/T-1 are made with the data for other systems. The difference in 1/T-1 between two Cu sites is discussed from the viewpoint of local inhomogeneous electronic states.