Anthropic filed paperwork to go public just as corporate America is entering its AI sticker shock phase.Why it matters: Companies are Anthropic's biggest customers. If they dial down their AI spend, that could weaken the AI lab's revenue just as the it prepares to IPO.Driving the news: Hours after Anthropic filed its pre-IPO paperwork, OpenAI CEO Sam Altman told CNBC that corporate concern over AI costs is "the most fair criticism of AI so far." Bain published a survey of nearly 1,000 companies showing that after investing in AI, "the value didn't arrive," with 40% of surveyed companies reporting AI cost savings below 10%.An early Anthropic investor tells Axios that companies are waking up to how much they're spending on Claude, Anthropic's AI model, and that is a risk they're monitoring closely. This comes after an AI consultant told Axios a CFO client accidentally spent half a billion dollars on Claude in a single month.Between the lines: Even AI executives are acknowledging their technology has a cost problem. "The risk of enterprises switching to cheaper models is existential and, frankly, escalating," Matt Rodgers, co-founder and CEO of Mill, who also worked on the original iPhone, told Axios via email. "Some open source LLMs [large language models] are as good without the price tag," he added.Threat level: Corporate pushback on AI spend would be a challenge for every AI lab, but Anthropic could feel it more given its exposure to enterprise customers.In April, Anthropic surpassed OpenAI in business customers for the first time, per Ramp data.Business revenue has been Anthropic's greatest strength, given these customers pay more than everyday people. It could become Anthropic's Achilles heel if businesses start to rebel against AI costs.Reality check: Anthropic is on track for nearly $50 billion in annual revenue per its latest funding round, and its first profitable quarter ever according to the Wall Street Journal. Anthropic keeps beating its own growth metrics, while competitor OpenAI is reportedly missing internal revenue targets.It's also the fastest growing company in modern American history.But the AI race is far from over: "You can't make a three or five year bet in this space... someone can jump over everybody else by coming up with the next great thing," Michael Levine, CFO of Fireblocks told Axios. The bottom line: AI labs are looking to go public right as their biggest customers are figuring out how to define their relationship with AI.