We consider an inhomogeneous linear Boltzmann equation in a low temperature regime, in the presence of an external force deriving from a single-well potential and with a collision operator featuring multiple conservation laws. We start by giving a description of the purely imaginary spectrum of the associated operator. We then go further and provide a hypocoercive result on the spectrum with real part smaller than $h$. It enables us to obtain some information on the long time behavior of the solutions and in particular to show the existence of metastable states. This is a joint work with Frédéric Hérau and Dorian Le Peutrec.