The actual colorbar range will be from 0 to the number of labels.
The actual colorbar ticks are at x.5 for each integer between the limits.
The colorbar tick labels are in the format you specified in the label vector.
% Create and set the colormap
RGB=[0.384 0 0
0.525 0 0
0.717 0 0
0.741 0.168 0
0.765 0.333 0
0.765 0.525 0
0.765 0.717 0
0.585 0.729 0.18
0.405 0.738 0.36
0.045 0.762 0.72
0.033 0.582 0.732
0.024 0.402 0.741
0.012 0.225 0.75
0.003 0.045 0.762
0 0 0.621
0 0 0.429];
colormap(RGB)
cb = colorbar();
% Set color labels (one for each row in RGB)
label = [ 2.7290e+000
4.2512e+000
6.6224e+000
1.0316e+001
1.6070e+001
2.5034e+001
3.8998e+001
6.0750e+001
9.4634e+001
1.4742e+002
2.2965e+002
3.5774e+002
5.5728e+002
8.6812e+002
1.3523e+003
2.1066e+003];
% Scale the colorbar and set the ytick labels.
% Since the numeric labels are nonlinear, we
% must use yticklabel; otherwise we could have
% just set them directly using ytick.
caxis([0,numel(label)])
cb.YTick = 0.5 : 1 : numel(label);
labelChar = strsplit(sprintf('%0.2e ',label));
cb.TickLabels = labelChar(1:end-1);
cb.FontSize = 8;
You may want the set the tick direction to out so the y ticks can be seen better.
cb.TickDirection = 'out'; % (not show in the image below)
