In general, businesses can charge what they like for their products
If the consumer does not like the price (or anything else) they are free to choose a different business or elect not to use the product.
A business cannot change a contracted price during the course of a contract except as provided for in that contract. For the particular service you are using the terms are here with info on price changes here in 2(c):
c. Pricing and payment. Payments are due and must be made according to the Offer Details for your Subscription.
(i) For Commitment Offerings, the price level may be based on the quantity of Online Services you ordered. Some offers may permit you to modify the quantity of Online Services ordered during the Term and your price level may be adjusted accordingly, but price level changes will not be retroactive. During the Term of your Subscription, prices for Online Services will not be increased, as to your Subscription, from those posted in the Portal at the time your Subscription became effective or was renewed, except where prices are identified as temporary in the Offer Details, or for Previews or Non-Microsoft Products. All prices are subject to change at the beginning of any Subscription renewal.
(ii) For Consumption Offerings, pricing is subject to change at any time upon notice.
Assuming your storage is a "Commitment Offering" the price cannot change during the course of the agreement but can be changed on renewal. If it's a "Consumption Offering" (pay-as-you-go), then it can be changed at any time.
Your response as a consumer is to either accept that price rise or stop using the service.
Assuming there are no laws in place, would class action lawsuits win if one showed that the costs of the provider had not increased?
No. As stated, businesses can charge what they like irrespective of the cost of production with the exception that they are generally prohibited from charging less than the marginal cost of production if by doing so they hurt other businesses (predatory pricing) or markets (anti-dumping).
Governments can and do intervene in markets in all sorts of ways for all sorts of reasons including setting prices and no market is completely unregulated. However, direct price controls are rather uncommon.