No, we can use less gas and pay the same. People waste plenty of energy in lots of ways.
Yes, we waste energy, but raise prices, especially hitting the lowest part of the society is definitively outrageous. Personally seeing the energy price rise I feel a small inflation in what I buy normally and have a confirmation of having made a good investment building a BBC house with a small p.v. BUT for my neighbor that's just means pay much more, not for the environment but for some humans profits.
If such raise are ONLY for "natural causes" that's would not be much fun, but still just, who consume more pollute more pay more. However present rises are NOT (at least largely not only) due to natural causes. Gas price and so electricity price in EU does not skyrocket because of the Russian but because EU Commission choose deliberately to negotiate them with monthly based future instead of long term contracts (look for Dutch TTF). The VERY SAME thing that happen in KZ: https://www.energyintel.com/download?issueId=0000017e-2c65-d... and who profit is very clear: https://oilprice.com/Energy/Crude-Oil/North-Sea-Oil-Operator... and https://www.theguardian.com/commentisfree/2022/jan/11/energy... also is clearly stated that such move is done to "rebalancing energy prices" (p. 148) https://assets.publishing.service.gov.uk/government/uploads/... for the Green New Deal purpose and NOT to make a Green New Deal "possible" but to make their private-public actors super-earn a very big amount of money. Normally that's have a name: embezzlement.
That's what taxes are there for: encourage purchases that serve the general interest.
To force social change it's better incentive the new, leaving the old as always, having a first wave of citizens that try and speak enthusiastically about the new, encourage public debate teaching why something should or must change etc.
For instance lying knowing that you are lying affirming that energy prices are skyrocketing because of the bad Russian bear is the ideal recipe to create social unrest, NOT to push a real green new deal. Clearly stating that some resources are more and more scarce, using the media power for that instead of to spread lies, is far more effective: peoples who eventually do not know and refuse to at first will get convinced, peoples with different interests can't just spot the lie weaponizing it against the change. I'm firmly convinced that talking bend society, trying to master citizens like cows in a herd, like Gustave Le Bon and Eduard Bernays have done in the past do work for a short period of time and the outcome are a series of straw fires followed by deep crisis, global wars, sufferance. While real Democracies have proved to be fragile but also an incredibly effective way to evolve and cooperate. Neoliberal style propaganda is useful and works not to master the society for the society sake but to pasture peoples as cows for élite sake...