  We study conformal field theories at finite temperature in the presence of a
temporal conformal line defect, wrapping the thermal circle, akin to a Polyakov
loop in gauge theories. Although several symmetries of the conformal group are
broken, the model can still be highly constrained from its features at
zero-temperature. In this work we show that the defect and bulk one and
two-point correlators can be written as functions of zero-temperature data and
thermal one-point functions (defect and bulk). The defect one-point functions
are new data and they are induced by thermal effects of the bulk. For this new
set of data we derive novel sum rules and establish a bootstrap problem for the
thermal defect one-point functions from the KMS condition. We also comment on
the behaviour of operators with large scaling dimensions. Additionally, we
relate the free energy and entropy density to the OPE data through the
one-point function of the stress-energy tensor. Our formalism is validated
through analytical computations in generalized free scalar field theory, and we
present new predictions for the O(N) model with a magnetic impurity in the
$\varepsilon$-expansion and the large N limit.
