  This paper studies convergence to equilibrium for the spatially inhomogeneous
linear relaxation Boltzmann equation in Boltzmann entropy and related entropy
functionals the $p$-entropies. Villani proved in \cite{V09} entropic
hypocoercivity for a class of PDEs in a H\"{o}rmander sum of squares form. It
was an open question to prove such a result for an operator which does not
share this form. We show exponentially fast convergence to equilibrium with
explicit rate in entropy for a linear relaxation Boltzmann equation. The key
new idea appearing in our proof is the use of a total derivative of the entropy
of a projection of our solution to compensate for additional error term which
appear when using non-linear entropies. We also extend the proofs for
hypocoercivity of both the linear relaxation Boltzmann and kinetic
Fokker-Planck to the case of $p$-entropy functionals.
