Energy consumption of mathematical operations
once a second

calculate/benchmar how much time that calculation takes. Wild guess: less than 200 CPU cycles. You didn't tell us anything about your MCU, but let's assume it runs at 1 MHz (that's far slower than modern MCUs usually run). Then, that's 2·10⁻⁴ s worth of power consumption. Does that matter to your application significantly? 
We can't tell you, because you don't tell us how much of a second your device is not in sleep. I bet it doesn't matter much.

The precision needed is only around four significant digits

Then use a cheap power series approximation. As you know,
$$e^x = \sum_{k = 0}^{\infty} \frac{x^k}{k!} = 1 + x + \frac{x^2}{2} + \frac{x^3}{6} + \frac{x^4}{24} + \cdots\text.$$
If you know the range of your \$x\$, then it's pretty easy to figure out how many elements of that sum you need to achieve your desired accuracy.
Also note that it's usual that you don't actually need \$e^x\$. Any other exponential base works too, and is just a factor away; i.e. you can use \$2^x\$, which many FPUs can give you quite trivially and energy efficient.
Other can't. We're not talking about &quot;slightly different amounts of energy here&quot;, we're talking about fundamentally different approaches to calculating the same.
So, conclusion:

Nope, there's no such rough number as you hope for
It really depends on your hardware, and your algorithmic choices.
once a second is ridiculously rarely, and unless you're not building something that is otherwise extraordinarily power-efficient, will not matter.