Carroll black holes in (A)dS spacetimes and their higher-derivative modifications

We define the Carrollian black holes corresponding to the limit of Schwarzschild--(anti)de Sitter (A)dS black holes and its higher-derivative counterpart known as Schwarzschild-Bach-(A)dS black holes, which is also a static spherically symmetric vacuum solution of quadratic gravity. By analyzing the motion of massive particles in these geometries, we found that, in the case of Schwarzschild-(A)dS black hole, a (nearly) tangential particle from infinity will wind around the extremal surface with a finite number of windings depending on the impact parameter and the cosmological constant. In Schwarzschild-Bach-(A)dS black hole, a particle passing close enough to the extremal surface will have an infinite number of windings; hence, it will not escape to asymptotic infinity as in Schwarzschild-(A)dS black hole. We also calculate the thermodynamical quantities for such black holes and argue that it is analogous to an incompressible thermodynamical system with divergent entropy when the temperature goes to zero (in the strict Carroll limit). We then define a divergent specific heat that can be positive, negative, or zero.