  Lurie and Gepner--Meier each define equivariant cohomology theories, namely
"tempered cohomology" and "equivariant elliptic cohomology," respectively,
using derived algebraic geometry. We construct a natural equivalence between
the equivariant elliptic cohomology of a preoriented abelian variety
$\mathsf{X}$ and the tempered cohomology of the associated preoriented
$\mathbf{P}$-divisible group $\mathsf{X}[\mathbf{P}^\infty]$. Moreover, we
emphasise the naturality and coherence of both these equivariant theories as
well as our comparison. To demonstrate the use of this comparison, we show that
equivariant topological modular forms $\mathrm{TMF}^{\mathbf{B} G}$ is
dualisable as a $\mathrm{TMF}$-module for all compact Lie groups $G$ which
decompose as a product of a torus and a finite group by formally reducing to an
argument of Gepner--Meier.
