Strictly speaking, the quoted claim is most immediately one about Hans Sloane's writings. So if this is an accurate description of what Sloane wrote, Olusoga is telling the truth. Olusoga is not specific about which writings he is referring to. At least some of Sloane's writings are relatively easy to find online, but so far, I'm not able to pinpoint quotes that fit what Olusoga attributes to him. In this sense, I would say the claim is unconfirmed, until we can identify the sources of specific quotes consistent with Olusoga's representation.
Ultimately the more germane question though is about British 17th-century slave plantations in general. It is easy to find systematic studies of mortality patterns on slave plantations such as this one about the US South and this one on Trinidad. These generally apply to the 18th and 19th centuries, rather than the 17th. They do show periods of high mortality, and some evidence to suggest malnutrition was at least a contributing cause.
An article about Sloane's "Natural History of Jamaica" has an interesting footnote:
Dunn states that while an estimated 12,000 Englishmen came to Jamaica in the first 6 years (1655-1661) of its settlement, by the end of that time the colony's population was only 3,470. Tropical fevers and starvation accounted for most of the deaths. Although the mortality rate had improved by the time Sloane came to Jamaica, the island still had the reputation of being a tropical hell.
If even a small number of white British settlers were starving, it would hardly be surprising if their slaves were too. In any event, here is a more directly relevant quote from Dunn's book, describing Barbados c. 1680:
In only one generation these planters had turned their small island into an amazingly effective sugar-production machine and had built a social structure to rival the tradition-encrusted hierarchy of old England. But the irony is that in accomplishing all this they had made their tropical paradise almost uninhabitable. By crowding so many black and white laborers on to a few square miles they had aggravated health hazards and over-taxed the food supply, condemning most inhabitants of the island to a semistarvation diet. Those who had money squandered it by overdressing, overeating, and overdrinking and by living in ornate English-style houses unsuited to the climate. Even the rich were unhappy in Barbados, for they suffered from claustrophobia, heat, and tropical fevers and longed for the dank, chill weather they were used to at home. Most of all they hated and feared the hordes of restive black captives they had surrounded themselves with. The mark of a successful Barbados planter was his ability to escape from the island and retire grandly to England. (p. 116)
Dunn gives various pieces of historical evidence to support his description. He cites a contemporary witness who described a particular planter (Edward Atcherley) as "an irresponsible drunkard who idled away his time at Port Royal while the neglected slaves and servants at Bybrook starved, stole, plotted, or ran away (p. 217)." Dunn points out that the a 1688 update to the Barbados Slave Act, "admitted that some Negroes stole food because they were starving." (p. 242) He clearly shows that the plantation called Bybrook had periods of high mortality and that an underfeeding of the slaves was a contributing factor.
Based on this I would say that yes, there is considerable historical basis for the statement, if we do not insist on taking it too literally. We don't know based on the above that it was cheaper to replace slaves then feed them because planters were clearly cruel and incompetent and may not have been making economically rational decisions. There may also be room to argue that death due to malnutrition in conjunction with other causes may not literally equal "starvation" but I think that point is rather trivial in this context.
EDIT: Several comments have pushed for direct data on the question of relative prices. According to Galenson in late 17th century Barbados, an adult male slave cost around £20 while a young girl might be closer to £10, sometimes less. Meanwhile, Eltis discusses some relevant estimates of yearly provisions for slaves and servants for around the same time, mostly in the range of £1 to £5 per year. Whether this data shows that underfeeding slaves fit some kind of rational cost-benefit analysis for some planters or not, I have absolutely no idea.