  Let $\mathbf{X}=\{X_t\}_{t\geq 0}$ be a L\'{e}vy process in $\mathbb{R}^d$
and $\Omega$ be an open subset of $\mathbb{R}^d$ with finite Lebesgue measure.
The quantity $H (t) = \int_{\Omega} \mathbb{P}^{x} (X_t\in \Omega ^c) d x$ is
called the heat content. In this article we consider its generalized version
$H_g^\mu (t) = \int_{\mathbb{R}^d}\mathbb{E}^{x} g(X_t)\mu( d x )$, where $g$
is a bounded function and $\mu$ a finite Borel measure. We study its asymptotic
behaviour at zero for various classes of L\'{e}vy processes.
