CO2 was a long-term consequence of the climate-biological system, being decoupled or even showing inverse trends with temperature

See CO2 and temperature decoupling at the million-year scale during the Cretaceous Greenhouse by in Nature Abel Barral, Bernard Gomez, François Fourel, Véronique Daviero-Gomez & Christophe Lécuyer. Here is the abstract:

"CO₂ is considered the main greenhouse gas involved in the current global warming and the primary driver of temperature throughout Earth’s history. However, the soundness of this relationship across time scales and during different climate states of the Earth remains uncertain. Here we explore how CO₂ and temperature are related in the framework of a Greenhouse climate state of the Earth. We reconstruct the long-term evolution of atmospheric CO₂ concentration (pCO₂) throughout the Cretaceous from the carbon isotope compositions of the fossil conifer Frenelopsis. We show that pCO₂ was in the range of ca. 150–650 ppm during the Barremian–Santonian interval, far less than what is usually considered for the mid Cretaceous. Comparison with available temperature records suggest that although CO₂ may have been a main driver of temperature and primary production at kyr or smaller scales, it was a long-term consequence of the climate-biological system, being decoupled or even showing inverse trends with temperature, at Myr scales. Our analysis indicates that the relationship between CO₂ and temperature is time scale-dependent at least during Greenhouse climate states of the Earth and that primary productivity is a key factor to consider in both past and future analyses of the climate system."