<p>The coupling of chiral superfluids to the geometry of their supporting surface leads to unexpected geometric induction effects. Yet, the influence of thermal fluctuations of this geometry remains largely unexplored. Here, we show that such fluctuations have significant consequences even in nearly flat substrates. Using the Ginzburg-Landau free energy for a chiral superfluid coupled to geometry, we find that shape fluctuations of the flexible substrate renormalize the vortex interaction strength. In the low-tension limit, this softens the interaction at large distances and lowers the Berezinskii-Kosterlitz-Thouless transition temperature. Our renormalization-group analysis of a dual sine-Gordon theory shows that, conversely, chiral superfluid order can suppress surface fluctuations, leading to an extended phase diagram that links the superfluid transition to the crumpling transition of the substrate. These reveal thermodynamic signatures of chirality and geometry in the superfluid transition.</p>