In this talk I will present a finite volume scheme for a cross-diffusion system which models chemotaxis with local sensing. This system has the same gradient flow structure as the celebrated minimal Keller-Segel system, but unlike the latter, its solutions are known to exist globally in 2D. The long-time behavior of solutions is only partially understood which motivates numerical exploration with a reliable numerical method. For this purpose, I will introduce a linearly implicit, two-point flux finite volume approximation of the system. I will explain why this scheme preserves, at the discrete level, the main features of the continuous system, namely mass, non-negativity of solution, entropy, and duality estimates. These properties will be crucial to prove the well-posedness, unconditional stability and convergence of the scheme. Moreover, this scheme possesses an asymptotic preserving (AP) property in the quasi-stationary limit. Finally, I will present some numerical experiments illustrating the convergence and AP properties of the scheme as well as its reliability with respect to stability properties of steady solutions. This work is in collaboration with M. Herda (INRIA Lille) and A. Trescases (Institut de Mathématiques de Toulouse).