The function y=x^p is "scale-invariant" in the following sense.  Consider an interval such as (x,2x), where y changes from x^p to  2^px^p.  Now scale x by a scale factor a.  We look at the interval (ax,2ax) where y changes from (ax)^p to 2^p(ax)^p.  We see that y in this interval is the same (except for a scale change of a^p) as y in the unscaled interval.

Most functions do not behave this way.  Consider y=exp(x), which goes [from exp(x) to exp(2x)] over the interval (x,2x), and [from exp(ax) to exp(2ax)] in the scaled interval (ax,2ax).  There is no scale factor that can be removed from y to make the second interval of y appear the same as the first.

The sum of two different powers is usually not scale invariant.  However, special cases may still be.  y=Ax² + Bx can be written as A(x+B/2A)² -B²/4A.  If the new variable X=x+B/2A is scaled to aX, then y(aX)=a²Y(X)+(a²-1)(B²/4A).  In other words, the function y(x) is scale invariant (with scaling exponent 2) after finding the right scaling variable X and allowing for a scale-dependent shift of y.  In this sense, the thermodynamic functions seem to display scale invariance around the critical point (for systems that have a critical point!), with the scaling variable t=(1-T/T_c).  Note that the logarithm y=log x obeys y(ax)=y(x) + log a.  It is scale invariant with exponent 0 (and a scale-dependent shift.)  This is related to the famous formula
    lim_p-->0 (x^p-1)/p = log x
which shows that logs are a special case of power law functions with power 0.