Yes, there is.
But the data is still very sparse and errors are large.
Past atmospherics pressures have been estimated by at least three different methods:
- Isotopic composition of fluid inclusions trapped hydrothermal quartz ( Nishizawa ey al 2007; Goldblatt et al 2009; Marty et al, 2013)
Image from Fig. 2 of Nishizawa et al 2007.
- Size distribution of gas bubbles in basaltic lava flows (Som et al 2016)
Figure 3c of Som et al (2016): Beasley River geologic context and flow detail (scale bar, 1 cm)
- Size distribution of fossilised raindrop imprints (Som et al, 2012; Kavanagh & Goldblat, 2015)
Figure 1 of Som et al (2012) The 2.7-billion-year-old Ventersdorp Supergroup raindrop imprints lithified in tuff at Omdraaivlei, South Africa.
Each study is based on samples that capture the conditions at a fairly specific point in time. Therefore, different results not necessarily contradict each other, but offer a sense of how variable has been the atmospheric pressure over geological time.
The size distribution of fossilised raindrop imprints might have a large range of error due to the many factors that influence drop size beside atmospheric pressure. However, some studies suggest that the Archean (4 to 2.5 billion years ago) atmosphere was almost ten times denser than it is today (Kavanagh & Goldblat, 2015). That figure is based on fossilized raindrop imprints dated about 2.7 billion year ago, that is after the evolution of photosynthesis, but still in its early stages, when most oxygen was absorbed by the oceans and there was very little of it in the atmosphere.
On the other hand, studies based in isotopic composition of fluid inclusions and bubbles in basaltic lava have found that the atmosphere was less dense than it is today.
These studies are of great interest, as a denser atmosphere even if poor in greenhouse gasses can produce warmer surface condition, helping to resolve the faint young Sun paradox. In a nutshell, this paradox refers to how could liquid water exist on Earth in the past when the sun was much fainter than it is today.
It is important to note that the additional temperature in a thick atmosphere doesn't come from adiabatic warming as some people have suggested here. A good treatment of the phenomena is presented by Chemke et al (2016) in the paper "The thermodynamic effect of atmospheric mass on early Earth's temperature", there they say:
We find that higher atmospheric mass tends to increase the
near-surface temperature mostly due to an increase in the heat
capacity of the atmosphere, which decreases the net radiative cooling
effect in the lower layers of the atmosphere. Additionally, the
vertical advection of heat by eddies decreases with increasing
atmospheric mass, resulting in further near-surface warming.