I was able to acquire access to a couple of reference tomes, and see what they had to say about this huge but mysterious temple. Here’s the first of them. Sadly the figure was not well reproduced in my copy.
From L. Richardson, A New Topographical Dictionary of Ancient Rome, 1992, 341-2:
Remains of a very large temple that faced east stood south of Montecavallo until the seventeenth century. Together with its stair, this extended from Piazza della Pilotta to the fountain of Montecavallo (Fig. 72). The rear corner of the temple, built of blocks of peperino and carrying the marble entablature and a corner of the pediment, against which was built a medieval defense tower, was known variously as Torre Mesa, Torre di Mecenate, and Frontispizio di Nerone. Remains of a great stair leading to the temple from the plain below still survive in the gardens of Palazzo Colonna and the Pontificia Universita Gregoriana, and records of these have been left by artists, notably van Heemskerck, who gives a panorama of what was to be seen in the sixteenth century (2.81 v, 82 r). There are also a plan by Palladio (Zorzi pls. 153-55 ) and drawings of the entablature and corner of the pediment by Serlio and the Anonymous Destailleur (RomMitt 52 : 95 fig. 1). Fragments of the architecture, including an architrave block, parts of the frieze, and the corner block of the pediment, still lie in the gardens of Palazzo Colonna.
This complex was the subject of a famous debate toward the end of the nineteenth century between Hulsen, who wished to identify it as the Temple of Serapis (see Serapis, Aedes), built by Caracalla, and Lanciani, who held it to be the Temple of Sol (see Sol, Templum) built by Aurelian. Each advanced relays of argument for his identification, and since then topographers have generally held for one theory or the other. Most recently Nash (2.376-83) and Lugli (Lugli 1938, 304-7 ) have sided with Hulsen, whereas M . Santangelo (MemPontAcc, ser. 3.5 [1940-41]: 154-77 ) has sided with Lanciani. Only H. Kàhler (RomMitt 52 : 9 4 —105) has been bold enough to reject both identifications, yet he is unquestionably correct. The architectural ornament of the temple is unmistakably Hadrianic (cf. PBSR, n.s., 8, vol. 21 : 118-51 [D. Strong]). Moreover, the pronaos, as Palladio has drawn it, is a close congener of the pronaos of Hadrian’s Pantheon, with its lines of columns leading back to important niches between pronaos and cella. It has been argued that the brickwork in the walls of the monumental stair approaching the temple is typically Severan (see Lugli 1938, 306-7), but there seems to have been no confirmation of this from the evidence of brickstamps. If it is Severan, it must be a later addition to a Hadrianic building.
Palladio shows the temple as peripteral, sine postico, pseudo-dipteral, with twelve columns on the façade and fourteen down the flanks. It is mounted on a platform with seven steps running around the three colonnaded sides. The pronaos is deep, with eight columns in pairs behind the third, fifth, eighth, and tenth columns of the façade. These flank niches in the cella wall, semicircular to either side, and rectangular for the door in the middle. The interior is believed to have been hypaethral, with colonnades down the sides in two storeys, Ionic below, Corinthian above. The total height of the main order has been calculated as 21.17 m (Alberti), the entablature as 4.83 m. It was a huge temple, on the order of the Temple of Venus et Roma, and set at the back of a large precinct finished, at least along the back, with a wall behind an addorsed colonnade, in the bays of which were niches, alternately rectangular and semicircular. At the front of the precinct were found the statues of the horse tamers that still adorn Montecavallo, although perhaps they belonged to the Thermae Constantinianae (MemPontAcc, ser. 3.5 [1940-41]: 158, 161 [M. Santangelo]).
The approach from the plain of the Campus Martius was complicated, and the drawings of it are difficult to read. It consisted of a double stair on each side of an open court, the inner stair on each side steeper than the outer. The stairs were roofed, so there was a subtle element of surprise introduced, but there were windows along the sides, so one could admire the view along the way. At the top one had to make a detour to enter the precinct, where the view of the flank of the temple would be enhanced. The stairs were carried on vaults, and a number of vaulted chambers filled the back of the court between them. It is not clear what the use of these rooms may have been. Lanciani (LS 1.38) believed that blocks of these stairs were robbed in 1348 to build the stair leading up to the church of S. Maria in Aracoeli.
The arguments in favor of identifying this as the Temple of Salus are simply that it is in approximately the right place with respect to the Porta Salutaris and would have a certain prominence, consonant with its having been repeatedly struck by lightning. We know of no Hadrianic rebuilding of the Temple of Salus, but coins bearing the image of Salus and the legend Salus Augusti are particularly numerous in Hadrian’s principate (see, e.g., B. M. Coins. Rom. Emp. 3. cxlviii-clxix).
RomMitt 52 (1937): 94-105 (H. Kahler); Lugli 1938, 304-7 ; MemPontAcc, ser. 3.5 (1940-41): 154-77 (M. Santangelo); PBSR, n.s., 8, vol. 21 (1953): 118-51 (D. E. Strong); Nash 2.376-83 ; M. A. Marwood, The Roman Cult of Salus (BAR, Int. Ser. 465 ): especially 2-15 .
[Lugli, G. I monumenti antichi di Romae suburbio. Vol. 3, A traverso le regioni. Rome 1938.
MemPontAcc = Memorie: Atti délla Pontificia Accademia Romana di Archeologia.
PBSR = Papers of the British School at Rome.
RomMitt = Mitteilungen des Deutschen Archäologischen Instituts, Römische Abteilung.]