In general, we know that adding a subset to a regular cardinal $\kappa$ can collapse cardinals. If, for example, there is $\gamma < \kappa$ with $2^\gamma >\kappa$, then $Add(\kappa,1)$ will collapse $2^\gamma$ to $\kappa$, since every subset of $\gamma$ will appear as a block in the generic.

However, this argument requires that we use the poset defined in $V$. We can easily construct situations in which $2^\gamma > \kappa$ but forcing with $Add(\kappa,1)^L$ does not collapse cardinals, by the same techniques that allow us to achieve Easton's Theorem. If we force over $L$ with $Add(\gamma, \kappa^+) \times Add(\kappa,1)$, for example, then cardinals are preserved. I am interested to know if this holds in general.

Does $Add(\kappa,1)^L$ ever collapse cardinals?