Foxton Beach resident Christina Paton said the district could not afford to provide so much money.
She said a previous financial indication from HDC was that the annual shortfall would be around $200,000.
In July, HDC received a $1 million grant from the Ministry for Arts, Culture and Heritage to go towards construction of the building.
A condition of that grant was that HDC would agree to fund reasonable maintenance, operating and staff costs for exhibition and gallery spaces.
Mrs Paton said the word 'reasonable' put a lot of liability on HDC to cover costs.
A former HDC councillor, Anne Hunt, said she was assured as a councillor when the museum's trust deed was first drawn up that council would not be liable for operating costs for the museum part of the facility.
"We need to be assured we are not going to have a run-away budget," Mrs Hunt, speaking as a member of the public, told the HDC meeting last month.
"I'm stunned and shocked," she said this week, after learning of the half million dollar shortfall.