The volume \( V \) of a rectangular pyramid is given by the formula:
\[ V = \frac{1}{3} \times L \times W \times H \]
where \( L \) is the length, \( W \) is the width, and \( H \) is the height.
For the original pyramid, we have:
- \( L = 24 \)
- \( W = 22 \)
- \( H = 28 \)
Calculating the volume of the original pyramid:
\[ V = \frac{1}{3} \times 24 \times 22 \times 28 \]
Calculating \( 24 \times 22 \):
\[ 24 \times 22 = 528 \]
Now, calculating \( 528 \times 28 \):
\[ 528 \times 28 = 14784 \]
Now, substituting this value into the volume formula:
\[ V = \frac{1}{3} \times 14784 = 4928 \]
Now that we have the volume of the original pyramid \( V = 4928 \), we need to find the volume of the similar pyramid with a scale factor of \( \frac{2}{3} \).
The volume of similar figures scales by the cube of the scale factor. Therefore, the volume of the new pyramid can be calculated as:
\[ V_{\text{new}} = V_{\text{original}} \times \left(\frac{2}{3}\right)^3 \]
Calculating \( \left(\frac{2}{3}\right)^3 \):
\[ \left(\frac{2}{3}\right)^3 = \frac{8}{27} \]
Now calculating the new volume:
\[ V_{\text{new}} = 4928 \times \frac{8}{27} \]
Calculating this expression:
\[ V_{\text{new}} = \frac{4928 \times 8}{27} = \frac{39424}{27} \approx 1457.59 \]
Thus, the volume of the new pyramid, rounded to two decimal places, is approximately:
\[ \boxed{1457.59} \]