For warm and cold days, it is used the so called warm (ISA+5, ISA+10, ISA+15, etc.) and cold (ISA-5, ISA-10, ISA-15, etc.), where the increments (decrements) represent the difference with respect to the 288.15 [K] of an average day.
Given the new \(T_0\), and given the same pressure at sea level (\(p_0\) does not change), the new density at sea level can be calculated. Then the ISA equation are obtained proceeding in the same manner.