  I show that the summed spectral energy distribution (SED) $L(\nu)$ of any
extended blackbody radiator will scale in a predictable way if all parts of the
body change in temperature by the same factor $X$, such that $ L^\prime(\nu) =
X^3L(\nu/X) $. This should for example apply to accretion disks around black
holes, where $X$ is relative accretion rate, or external heating rate, but will
not apply to changes in black hole mass. I summarise evidence that AGN
optical-UV SEDs become progressively redder with decreasing luminosity, and
show that the trend in colour versus luminosity shown by Mushotzky and Wandel
(1989) is matched extremely well by taking a template high-luminosity SED and
scaling it in the manner described above. This agreement is striking because it
involves {\em no adjustable parameters}. The agreement breaks down at low
luminosities because of stellar contamination and reddening. I then consider
the colour changes of an individual AGN (NGC 5548) during luminosity changes,
which according to the popular X-ray reprocessing model, should follow the
scaling law well. However the observed changes are clearly {\em not} consistent
with the simple scaling prediction. Instead, these colour changes are quite
well explained by the mixing of a constant red component and variable blue
component. Overall, there is then strong support for the ideas (i) that AGN
optical-UV SEDs arise from accretion discs, (ii) that accretion rate plays a
significant role in the very large range of luminosity seen in AGN, and (ii)
that the inner regions of AGN vary independently of the outer accretion disc.
