Write a Python function `cube_sum_of_evens(n)` that calculates the sum of the cubes of the first `n` even natural numbers. The function should take an integer `n` as input and return the computed sum.
#### Example Usage
```python [main.nopy]
cube_sum_of_evens(2) # Output: 72
cube_sum_of_evens(3) # Output: 288
cube_sum_of_evens(4) # Output: 800
```
#### Explanation
- The first 2 even natural numbers are 2 and 4. Their cubes are 8 and 64, and their sum is 72.
- The first 3 even natural numbers are 2, 4, and 6. Their cubes are 8, 64, and 216, and their sum is 288.