  Supersymmetric partition function of $\mathcal N=1$ superconformal theories
on $S^1_{\beta} \times S^3$ is related to the superconformal index receiving
contributions from short representations. The leading coefficients in the small
$\beta$ (high "temperature") expansion of the index were previously related to
the conformal anomaly coefficients of the theory. Assumptions underlying
universality of these relations were tested only for simplest low-spin unitary
multiplets. Here we consider examples of higher derivative non-unitary
$\mathcal N=1$ multiplets that naturally appear in the context of extended
conformal supergravities and compute their superconformal index. We compare the
coefficients in the small $\beta$ expansion of the index with those proposed
earlier for unitary multiplets and suggest some modifications that should apply
universally to all types of theories. We also comment on the structure of
subleading terms and the case of $\mathcal N=4$ conformal supergravity.
