measurement level of exponential temperature in degrees Celsius

#1
I was wondering about the following theoretical example. Assume that I have measured temperature in degrees Celsius on various days (x), and for some strange reason converted each of these using e^x.
My question is now, what measurement level is this 'exponential temperature'?

My own thought would be to classify this is as ordinal and not as interval. Although temperature in Celsius is an interval scale, by taking the exponential of this, the scale is no longer linear. Therefor differences between two values are not comparable anymore. The difference between 4 and 2 degrees Celsius is the same as the difference between 8 and 6 degrees Celsius, but this is no longer the case if we look at the exponential temperature. A difference between 4 and 2 exponential temperature is not the same as between 8 and 6 exponential.

I guess my question boils down to asking if only linear scales are acceptable interval scales?

The article from Stevens (1946) suggests:
ordinal x' = f(x)
interval x' = ax + b
(p. 678)

If I read this right, it supports my idea.

Am I overlooking something?

Thanks in advance for any reply.