A local sensing Keller-Segel model, in which cells adjust their motility to the local chemoattractant rather than its gradient, and a semi-implicit finite volume scheme for its numerical approximation are studied. Computable a posteriori bounds for the approximation error measured in terms of the L∞(0,T;H−1(Ω)) and L2(0,T;L2(Ω)) norms of the cell density are derived based on a new stability estimate and suitable reconstructions of the flux and the flux potential of the numerical solution. Optimal scaling of the estimates is numerically demonstrated in 1D and 2D experiments. Moreover, an adaptive mesh refinement method steered by the estimator is proposed for the 1D case, and its efficiency is numerically verified.