Preheat the grill for medium-high heat.
In a medium saucepan, heat the oil over medium heat. Add the garlic and saute for 1-2 minutes, until fragrant. Stir in the cola, ketchup, vinegar, Worcestershire sauce and chili powder.
Bring to a boil, then reduce heat and simmer for about 15 minutes or until sauce reduces and thickens. Remove 6 tablespoons of sauce and set aside.
In a small bowl, combine brown sugar, paprika, salt, pepper, and cumin. Rub the spice mixture over the chicken. Place chicken on grill and baste with 3 tablespoons of the reserved sauce.
Cook for 5 minutes, then turn chicken over and baste with the remaining 3 tablespoons of reserved sauce. Cook for an additional 5 minutes, or until chicken is cooked through.
Serve with remaining sauce.