* I think the statement is that for any dimensional CFT the following is true,


$$\langle T^{\mu}\_\mu \rangle = \sum B\_n I\_n - 2(-1)^{d/2}AE\_d,$$


where $E\_d$ is the `"Euler density" and $I\_n$ are the independent "Weyl invariants of weight $-d$".


(...I am not sure of the definition of the geometric quantities coming on the R.H.S and I wonder if the notion of the "Euler density" and ``Weyl invariants" are related to the ideas of the Weyl tensor and the Euler tensor..)




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* For the special case of $3+1$ CFTs I am told that there is a relationship between the effective action and these anomaly coefficients as,


$W = \frac {a\_0}{\epsilon^4} + \frac {a\_1}{\epsilon^2} + a\_2 ln (\epsilon) + w(g)$


where I guess $\epsilon$ is the UV regulator, $W$ is I guess the connected functional/free energy defined as $W = - ln (Z)$ ($Z$ being the partition function), $w(g)$ is the UV finite part dependent on the metric $g$ such that $w(\lambda^2 g) = w(g) - a\_2ln(\lambda)$.


The most important and universal part of this answer is apparently that $a\_2$ is universal and has the following properties that, 


(1) $a\_2(e^{-2\omega}g) = a\_2(g)$


(2) $a\_2 = AE\_4 + BI\_4
$ where $E\_4 = \frac{1}{64}\int (R\_{\alpha \beta \mu \nu}R^{\alpha \beta \mu \nu} - 4R\_{\mu \nu}R^{\mu \nu} + R^2)$ and $I\_4 = -\frac{1}{64}\int (R\_{\alpha \beta \mu \nu}R^{\alpha \beta \mu \nu} - 2R\_{\mu \nu}R^{\mu \nu} + \frac{R^2}{3})$


Thus one says that the universal part of the free energy of a $3+1$ CFT is given by the integrated conformal anomaly. 


I would like to know what is the proof of the above and its generalization to higher dimensions (if known!). 




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It seems very mysterious to me that anomalies should determine the universal parts of a partition function! 



Proof of the general expression for anomaly in a CFT and its partition function
The answer to your question is contained in the recent book by Spyros Alexakis ``The Decomposition of Global Conformal Invariants'' published last year by Princeton University Press. He answers the Deser-Schwimmer conjecture which (as far as I can understand your question) is the broader context of what you are asking.


