**Strictly speaking, the quoted claim is most immediately one about [Hans Sloane's](https://en.wikipedia.org/wiki/Hans_Sloane) 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 fin systematic studies of mortality patterns on slave plantations such as this one about [the US South](https://sci-hub.tw/https://www.jstor.org/stable/1170958?seq=1) and this one on [Trinidad](https://sci-hub.tw/https://www.jstor.org/stable/2174434?seq=1). 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"](https://sci-hub.tw/https://www.jstor.org/stable/2674358?seq=1) has an interesting footnote:

> [Dunn](https://books.google.com/books/about/Sugar_and_Slaves.html?id=-7InAAAAYAAJ) 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](https://en.wikipedia.org/wiki/Barbados_Slave_Code),  "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.