Solution:

Since both gases \( A \) and \( B \) have the same temperature \( T \), pressure \( P \), and volume \( V \), each gas has the same number of moles, \( n \), by the ideal gas law:

\[
PV = nRT \Rightarrow n = \frac{PV}{RT}
\]

When gases \( A \) and \( B \) are mixed, the total number of moles becomes \( 2n \). Since temperature \( T \) and volume \( V \) remain the same, the total pressure \( P_{\text{mixture}} \) will be:

\[
P_{\text{mixture}} V = (2n)RT
\]

\[
P_{\text{mixture}} = 2P
\]

Answer: The pressure of the mixture is \( 2P \).