How to calculate Average temperature from area under the curve?

Let's call the horizontal axis m and the vertical axis T.

Consider a very small slice of the temperature curve starting from the time m to time m+dm. The area of this slice is $A= dm*T$, and its center of geometry is at the $T_{cg}=T/2$.

Then the average temperature of the 1 cycle is, just using the CG of the different segments and use the following eq i:

$$ T_{average} =frac {Sigma(A_n *L_{cgn})}{A_{total}}$$

You already mentioned in the question that you want is average, which is mean and not RMS value, use RMS values if the negative sign needs to be treated as positive temperature only.

Here you simply need to calculate the area under the curve (with sign), cummulate it and divide it by the total time period.

Will it be simple mean of the curve or rms of the curve?

The average temperature will involve a simple mean of the curve multiplied by the total time span of the curve.

If you have sampled data at a constant rate (i.e. the time $dt$ between data points is constant), then the average of the temperature will simply be the average of all the temperatures.