The present invention relates to a cross-linkable or cross-linked rubber composition having improved hysteresis properties in the cross-linked state which is useful for the manufacture of treads for tires. The rubber composition of the present invention comprises at least one diene elastomer having a molar ratio of units originating from conjugated dienes which is greater than 30% and comprising carboxylic acid functions along its chain, and a reinforcing inorganic filler. The present invention further relates to a process for preparing such a cross-linkable or cross-linked rubber composition as well as to a tire having reduced rolling resistance and a tread for a tire where the tread comprises the rubber composition of the present invention.