  When a (super) conformal field theory is placed on a non-trivial manifold,
the (super) conformal symmetry is broken. However, it is still possible to
derive broken Ward identities for these broken symmetries, which provide
additional constraints on the theory. We derive and apply the broken Ward
identities associated with the (super) conformal group on the thermal manifold
$\mathcal{M}_\beta = S_\beta^1 \times \mathbb{R}^{d-1}$ and $\mathcal{M} = T^2
\times \mathbb{R}^{d-2}$. The novel constraints not only systematically
reproduce known results, including an implicit formulation of the generalized
Cardy formula, but also elegantly relate the thermal energy spectrum with the
conformal spectrum.
