  Following the recent observation that the Ricci flow and the infinite
distance swampland conjecture are closely related to each other, we will
investigate in this paper geometric flow equations for AdS space-time
geometries. First, we consider the so called Yamabe and Ricci-Bourguignon flows
and we show that these two flows - in contrast to the Ricci flow - lead to
infinite distance fixed points for product spaces like $AdS_d\times S^p$, where
$AdS_d$ denotes d-dimensional AdS space and $S^p$ corresponds to a
p-dimensional sphere. Second, we consider black hole geometries in AdS space
time geometries and their behaviour under the Yamabe and Ricci-Bourguignon
flows. Specifically we will examine if and how the AdS black holes will undergo
a Hawking-Page phase transition under the Ricci flow, the Yamabe flow and under
the general Ricci-Bourguignon flow.
