We establish formulae for the derivative of the Poincaré constant of Gibbs measures on both compact domains and all of Rd. As an application, we show that if the (not necessarily convex) Hamiltonian is an increasing function, then the Poincaré constant is strictly decreasing in the inverse temperature, and vice versa. Applying this result to the O(2)model allows us to give a sharpened upper bound on its Poincaré constant. We further show that this model exhibits a qualitatively different zero-temperature behavior of the Poincaré and Log-Sobolev constants.