The floating area was previously investigated as a natural extension of classical affine surface area to non-Euclidean convex bodies in spaces of constant positive curvature. We introduce the family of $L_p$-floating areas for spherical convex bodies, as an analog to $L_p$-affine surface area measures from Euclidean geometry. We investigate a duality formula, monotonicity and isoperimetric inequalities for this new family of curvature measures on spherical convex bodies. Furthermore, using the $L_p$-floating area, we introduce a new entropy functional for spherical convex bodies and a dual isoperimetric inequality is established. Based on joint works with Florian Besau.