  We consider a free $p$-form gauge theory on a $d$-dimensional sphere of
radius $R$ and calculate its free energy. We perform the calculation for
generic values of $p$ and obtain the free energy as a function of $d, p$ and
$R$. The result contains a $\log R$ term with coefficient proportional to
$\left(2p+2-d \right)$, which is consistent with lack of conformal invariance
for $p$ form theories in dimensions other than $2p+2$. We also compare the
result for $p$-form and $(d-p-2)$-form theory which are classically Hodge dual
to each other in $d$ dimensions and find that they agree for odd values of $d$.
Instead, for even $d$, we find that the results disagree by an amount that is
consistent with the reported values in the literature.
