Mitigating Side Channel Attack (SCA)
As you are always decrypting the same flash content with the same key, statistical attacks like DPA do not work, as they try to find dependencies between two random variables (one being an intermediate value predicted using parts of the key and the other one being the measured current consumption(s)), where in your case one of them is constant.

The only uses an attacker can get from several measurements (instead of using just one) is to take the average to reduce the noise or to apply an attack that exploits the noise. Otherwise the attacker basically has to use an attack that works with a single power trace.

If an attacker is able to run the identical software on an identical device where (s)he can choose the key and the cipher text, (s)he can use it to run the profiling phase for a template attack, and hope that both devices have very similar power characteristics.

As template attacks calculate likelihoods depending on the observed noise, you can combine the results of several measurements to improve your chances to recover key and data.

From my experience the success probability for a very skilled attacker (using an optimal template attack implementation combined with key enumeration techniques) are relatively low if the AES is implemented in hardware, but pretty good for a software implementation on a leaky device.