At least to me, it's not immediately obvious how a recession might affect energy efficiency--which can be defined as the amount of energy needed to produce a given amount of output. A overall rise in energy efficiency is a consistent pattern over over time: for example, here's a figure showing US energy consumption divided by real GDP over the last 70 years or so. 
There are a variety of reasons for this long-run pattern. Developed economies over time tend to grow more slowly in energy-intensive industries like manufacturing and more quickly in service industries. As an environmental protection measure, governments often push for energy efficiency standards for everything from cars to buildings to appliances and electrical equipment. Companies that use a lot of energy have direct incentives to find ways to produce with less. In the US, the greater growth of population in warmer-weather states has also tended to reduce the growth of energy demand.
But what happens in a recession? Both economic output and energy use are likely to drop, but which one is likely to drop more--and thus how will energy efficiency be affected? The International Energy Agency has published its Energy Efficiency 2020 report (December 2020, free registration required). (The IEA is an autonomous Paris-based intergovernmental organization, somewhat similar to the OECD, which publishes a steady stream of energy-related reports.) The IEA has been warning for the last couple of years that from a global perspective, gains in energy efficiency have been declining, and the recession seems likely to worsen this situation.
Overall, the IEA expects global primary energy demand in 2020 to decrease by 5.3% from 2019. With global GDP falling by 4.6%, primary energy intensity improvement is projected to increase by only 0.8%, the lowest rate since just after the last global economic crisis in 2010 ... roughly half the rates, corrected for weather, for 2019 (1.6%) and 2018 (1.5%). This is well below the level needed to achieve global climate and sustainability goals. ... It is especially worrying because energy efficiency delivers more than 40% of the reduction in energy-related greenhouse gas emissions over the next 20 years in the IEA’s Sustainable Development Scenario ... This is well below the average annual improvement of more than 3% which would be consistent with meeting international climate and sustainability goals.
Here's some global data from the last couple of decades. As you can see, gains in energy efficiency fell during previous global recession, and remained low for a year (in 2010) after the economic recovery had started.
The IES report also has some interesting comments on how the pandemic recession is scrambling previous patterns of energy demand and shifts in energy efficiency. In the buildings sector, for example:
The buildings sector is witnessing a partial shift in energy demand from commercial to residential buildings, as social distancing and teleworking reduce use of commercial buildings and increase activities that use energy in the home. In the first half of 2020, electricity use in residential buildings in some countries grew by 20% to 30% while falling by around 10% in commercial buildings. In commercial buildings, essential services are accounting for a larger share of energy use. These services are often more energy-intensive, so the energy intensity of commercial buildings is likely to increase. For example, food sales outlets, which have largely continued to operate during the pandemic, are more than twice as energy-intensive as the average office in the United States, where many offices have been largely unoccupied during the crisis.
As shops and offices re-open, commercial buildings could become more energy intensive if occupants expect higher ventilation rates to reduce the risk of Covid-19 transmission. Around 30% of a building’s energy is dissipated in ventilation and exfiltration. This would only increase with higher ventilation rates. ...
The transportation sector is seeing a shift across modes:
Long-distance transport is witnessing dramatic falls in activity across all modes, with commercial aviation likely to be 60% lower in 2020 and rail demand 30% lower. The difference between these drops suggests that, at least domestically, some switching from planes to trains and cars is taking place. Shifts from aviation to rail would reduce energy intensity whereas a shift to road vehicles may increase energy intensity. In cities, people are moving away from public transport, which is down 50% in some countries, to private cars and active modes of transport such as walking, cycling or using other non-motorised vehicles.
A bright spot for energy efficiency is that many households are updating their appliances during the pandemic, and newer appliances tend to be more energy-efficient than those they replace:
A bright spot for technical efficiency gains is the appliances sub-sector. Data through the end of the third quarter of 2020 indicate that the Covid-19 crisis has increased households’ interest in new appliance purchases, with at least some appliances replacing older, inefficient models. Since the pandemic began, online shopping search indices were up by 20% to 40% for many appliance types worldwide, indicating that sales of appliances could be higher than usual. If these trends are confirmed, they would increase the technical efficiency of the global appliances stock.
But overall, the financial stresses of a pandemic recession are not a good time for investments in greater energy-efficiency--which may also be a reason why slower gains energy efficiency may persist even after a recession.
Investments in new energy-efficient buildings, equipment and vehicles are expected to decline in 2020, as economic growth falls by an estimated 4.6% and income uncertainty affects consumer and business decision making. Sales of new cars are expected to fall by more than 10% from 2019, keeping the overall vehicle stock older and less efficient, although the share of electric vehicles in new car sales is anticipated to grow to 3.2%, up from 2.5% in 2019.
Bottom line: If you are assuming that ongoing steady growth in energy efficiency will play a big role in meeting future environmental goals related to using fossil fuels, both standard air pollution goals and issue of carbon emissions, you should have already been worried by the trend to smaller annual gains in energy efficiency before the pandemic recession--and even more concerned now.