The approximate proportional relationship between cumulative carbon emissions and instantaneous global temperature rise (the carbon budget approximation) has proven to be a useful concept to translate policy-relevant temperature objectives into CO2 emissions pathways. However, when non-CO2 forcing is changing along with CO2 forcing, errors in the approximation increases. Using the GCAM model to produce an ensemble of ∼3000 scenarios, we show that linked changes in CO2 forcing, aerosol forcing, and non-CO2 greenhouse gas (GHG) forcing lead to an increase in total non-CO2 forcing over the 21st century across mitigation scenarios. This increase causes the relationship between instantaneous temperature and cumulative CO2 emissions to become more complex than the proportional approximation often assumed, particularly for low temperature objectives such as 1.5 °C. The same linked changes in emissions also contribute to a near-term increase in aerosol forcing that effectively places a limit on how low peak temperature could be constrained through GHG mitigation alone. In particular, we find that 23% of scenarios that include CCS (but only 1% of scenarios that do not include CCS) achieve a temperature objective of 1.5 °C without temperature overshoot.