Let $A$ be a semistable real matrix (*i.e.* the real parts of all the eigenvalues of $A$ are nonnegative). Let $P$ be a positive definite matrix.

Is it always true that $\text{trace}{A^{T}P+PA} \leq 0$?

P.S. In fact, slightly more is known about $A$: that $A+A^{T}$ is negative semidefinite.